. 


LIB 


LECTURES 


ON   THE 


THEORY  OF  ELLIPTIC 
FUNCTIONS 


BY 

HARRIS   HANCOCK 

PH.D.  (BERLIN),  DB.  Sc.  (PARIS),  PROFESSOR  OF  MATHEMATICS 
IN  THE  UNIVERSITY  OF  CINCINNATI 


VOLUME  I 

ANALYSIS 


FIRST    EDITION 

FIRST    THOUSAND 


NEW   YORK 

JOHN    WILEY    &   SONS 

LONDON:    CHAPMAN    &    HALL,    LIMITED 

1910 


- 


COPYRIGHT,   1910, 

BY 
HARRIS    HANCOCK 


Stanbope  ipress 

H.    GILSON     COMPANV 
BOSTON.     U.S. A 


GENERAL   PREFACE 


IN  the  publication  of  these  lectures,  it  is  proposed  to  present  the  Theory 
of  Elliptic  Functions  in  three  volumes,  which  are  to  include  in  general 
the  following  three  phases  of  the  subject: 

I.     Analysis; 
II.     Applications  to  Problems  in  Geometry  and  Mechanics; 

III.     General  Arithmetic  and  Higher  Algebra. 

In  Volume  I  an  attempt  is  made  to  give  the  essential  principles  of  the 
theory.  The  elliptic  functions  considered  as  the  inverse  of  the  elliptic 
integrals  have  their  origin  in  the  immortal  works  of  Abel  and  Jacobi.  I 
have  wished  to  treat  from  a  philosophic,  as  well  as  from  a  formal  stand 
point,  the  existence,  and  as  far  as  possible,  the  ultimate  meaning  of  the 
functions  introduced  by  these  mathematicians,  to  discuss  the  theories 
which  originated  with  them,  to  follow  their  development,  and  to  extend 
as  far  as  possible  the  principles  which  they  established.  In  this  develop 
ment  great  assistance  has  been  rendered  by  the  works  of  Hermite,  who 
contributed  so  much  not  only  to  the  theory  of  elliptic  functions  but  also  to 
almost  every  form  of  mathematical  thought.  The  theory  of  Weierstrass  is 
studied  side  by  side  with  the  older  theory,  and  the  beautiful  formulas  which 
we  owe  to  him  are  contrasted  with  the  corresponding  formulas  of  the 
earlier  writers.  Riemann  introduced  certain  surfaces  upon  which  he 
represented  algebraic  integrals,  and  by  thus  expressing  his  conceptions  of 
analytic  functions  he  revealed  a  clearer  insight  into  their  meaning. 
Instead  of  generalizing  either  the  theory  of  Jacobi  or  that  of  Weierstrass 
so  as  to  embrace  the  whole  subject,  it  is  thought  better  to  make  these 
theories  specializations  of  a  more  general  theory.  This  general  theory  is 
treated  by  means  of  the  Riemann  surface,  which  at  the  same  time  shows 
the  intimate  relation  between  the  two  theories  just  mentioned. 

In  Volume  II  a  treatment  of  elliptic  integrals  is  given.  Here  much 
attention  is  paid  to  the  work  of  Legendre,  whom  we  may  rightly  regard 
as  the  founder  of  the  elliptic  functions,  for  upon  his  investigations  were 
established  the  theories  of  Abel  and  Jacobi,  and  indeed,  in  the  very  form 
given  by  Legendre.  Abel  in  a  published  letter  to  Legendre  wrote:  "Si  je 
suis  assez  heureux  pour  faire  quelques  decouvertes,  je  les  attribuerai  a  vous 
plutot  qu'  a  moi  ";  and  Jacobi  wrote  as  follows  to  the  genial  Legendre: 
"Quelle  satisfaction  pour  moi  que  rhomme  que  j'admirais  tant  en 

iii 

781472 


IV  THEORY    OF   ELLIPTIC    FUNCTIONS. 

de"vorant  ses  ecrits  a  bien  voulu  accueillir  mes  travaux  avec  une  bonte*  si 
rare  et  si  precieuse!  Tout  en  manquant  de  paroles  qui  soient  de  dignes 
interpretes  de  mes  sentiments,  je  n'y  saurai  reprondre  qu'en  redoublant 
mes  efforts  a  pousser  plus  loin  les  belles  theories  dont  vous  etes  le  createur." 

True  Fagnano,  Euler,  Landen,  Lagrange,  and  possibly  others  had  dis 
covered  certain  theorems  which  proved  fundamental  in  the  future  develop 
ment  of  the  elliptic  functions;  but  by  the  patient  devotion  of  a  long  life 
to  these  functions,  Legendre  systematized  an  independent  theory  in  that 
he  reduced  all  integrals  which  contain  no  other  irrationality  than  the 
square  root  of  an  expression  of  degree  not  higher  than  the  fourth  into  three 
canonical  forms  of  essentially  different  character.  Thus  he  was  enabled 
to  discover  many  of  their  most  important  properties  and  to  overcome  great 
difficulties,  which  with  the  means  then  at  hand  appear  almost  insurmount 
able.  Methods  were  devised  which  furnished  immediate  results  and 
which,  extended  by  subsequent  investigations,  enriched  the  science  of 
mathematics  and  the  fields  of  knowledge.  In  this  direction  the  great 
English  mathematician  Cayley  has  done  much  work,  and  to  him  a  con 
siderable  portion  of  this  volume  is  due.  The  admirable  work  of  Greenhill 
has  also  been  of  great  assistance.  Much  space  is  given  in  Volume  II  to  the 
applications  of  the  theory.  These  applications  are  usually  in  the  form  of 
integrals  and  the  results  required  are  real  quantities,  and  for  the  most 
part  the  variables  must  be  taken  real.  Thus  the  complex  variable  of 
Volume  I  must  be  limited  to  some  extent  in  the  second  volume.  The 
problems  selected  serve  to  illustrate  the  different  phases  treated  in  the 
previous  theory;  sometimes  preference,  as  the  occasion  warrants,  is  given 
to  Legendre's  formulas,  sometimes  to  those  of  Weierstrass.  While  the 
most  of  these  problems  are  taken  from  geometry,  physics,  and  mechanics, 
there  are  some  which  have  to  do  with  algebra  and  the  theory  of  numbers. 

All  true  students  of  applied  mathematics,  engineers,  and  physicists  should 
have  some  knowledge  of  elliptic  functions;  at  the  same  time  it  must  be 
recognized  that  one  cannot  do  all  things,  and  it  is  not  expected  that  such 
students  should  be  as  well  versed  in  the  theoretical  side  of  this  subject  as 
are  pure  mathematicians.  For  this  reason  Volume  II  has  been  so  pre 
pared  that  without  dwelling  too  long  upon  the  intrinsic  meaning  of  the 
subject,  one  may  obtain  a  practical  idea  of  the  formulas.  Much  of  the 
theory  of  Volume  I  is  therefore  not  presupposed,  and  many  of  the  results 
that  have  hitherto  been  derived  are  again  deduced  in  Volume  II  by  other 
methods,  which,  without  emphasizing  the  theoretical  significance,  are 
often  more  direct.  This  is  especially  true  of  the  addition-theorems.  A 
table  of  elliptic  integrals  of  the  first  and  second  kinds  will  be  found  at  the 
end  of  this  volume,  which  may  consequently,  for  the  reasons  stated,  be 
regarded  as  an  advanced  calculus. 

Volume  III  will  be  of  interest  especially  to  the  lovers  of  pure  mathe- 


GENERAL   PREFACE.  V 

matics.  In  this  volume  the  theory  becomes  more  abstract.  Many 
problems  of  higher  algebra  occur  which  lie  within  the  realms  of  general 
arithmetic.  This  includes  the  theories  of  complex  multiplication;  of 
the  division  and  transformation  of  the  elliptic  functions;  a  study  of  the 
modular  equations  and  the  solution  of  the  algebraic  equation  of  the  fifth 
degree,  etc. 

The  discoveries  of  Kronecker  in  the  theory  of  the  complex  multiplica 
tion  not  only  prove  the  theorems  left  in  fragmentary  form  by  Abel  and  give 
a  clear  insight  into  them,  but  they  show  the  close  relationship  of  this 
theory  with  algebra  and  the  theory  of  numbers.  •  The  problem  of  division 
resolves  itself  into  the  solution  of  algebraic  equations,  and  the  introduc 
tion  of  the  roots  of  these  equations  into  the  ordinary  realm  of  rationality 
forms  a  "  realm  of  algebraic  numbers  ";  the  same  is  true  of  the  modular 
equations.  Kronecker,  Dedekind,  Hermite,  Weber,  Joubert,  Brioschi, 
and  other  mathematicians  have  developed  these  lines  of  thought  into  an 
independent  branch  of  mathematics  which  in  its  further  growth  is  sus 
ceptible  of  extension  in  many  directions,  notably  to  the  treatment  of  the 
Abelian  transcendents  on  the  one  hand  and  of  the  modular  systems  on 
the  other. 

Jacobi  in  a  letter  to  Crelle  wrote:  "  You  see  the  theory  [of  elliptic  func 
tions]  is  a  vast  subject  of  research,  which  in  the  course  of  its  development 
embraces  almost  all  algebra,  the  theory  of  definite  integrals,  and  the 
science  of  numbers."  It  is  also  true  that  when  a  discovery  is  made  in  any 
one  of  these  fields  the  domains  of  the  others  are  also  thereby  extended. 


INTRODUCTION   TO   VOLUME   I 


EVERY  one-valued  analytic  function  which  has  an  algebraic  addition- 
theorem  is  an  elliptic  function  or  a  limiting  case  of  one.  The  existence, 
formation,  and  treatment  of  the  elliptic  functions  as  thus  defined  are 
given  in  Chapters  I- VII  of  the  present  volume. 

An  algebraic  equation  connecting  the  function  and  its  derivative,  which 
we  have  called  the  eliminant  equation,  is  emphasized.  This  differential 
equation  due  to  Meray  is  first  used  as  a  latent  test  to  ascertain  whether 
or  not  a  function  in  reality  has  an  algebraic  addition-theorem,  and,  sec 
ondly,  as  shown  by  Hermite,  its  integrals  when  restricted  to  one-valued 
functions  are  one  or  the  other  of  the  three  classes  of  functions:  rational 
functions,  simply  periodic  functions,  or  doubly  periodic  functions.  We 
regard  the  first  two  types  as  limiting  cases  of  the  third,  the  three  types 
forming  the  general  subject  of  elliptic  functions.  All  three  types  of 
functions  are  shown  to  have  algebraic  addition-theorems,  and  conse 
quently  the  existence  of  the  eliminant  equation  is  found  to  be  coextensive 
with  that  of  the  elliptic  functions. 

In  Chapter  I  some  preliminary  notions  are  given.  In  particular  it  is 
found  that  the  rational  and  the  trigonometric,  and  later,  in  Chapter  V, 
that  the  doubly  periodic  functions  may  be  expressed  in  terms  of  simple 
elements,  and  it  is  seen  that  all  three  forms  of  expression  are  the  same; 
a  treatment  is  given  of  infinite  products  and  also  of  the  primary  factors 
of  an  integral  transcendental  function;  analytic  functions  are  defined. 

The  properties  of  functions  which  have  algebraic  addition-theorems 
are  considered  in  Chapter  II,  and  it  is  shown  that  these  properties  exist 
for  the  whole  region  in  which  the  function  has  a  meaning. 

After  establishing  the  existence  of  the  simply  and  doubly  periodic  func 
tions  in  Chapters  III  and  IV  and  after  studying  the  nature  of  the  periods, 
we  proceed  in  Chapter  V  to  the  actual  formation  of  the  doubly  periodic 
functions.  It  is  shown  that  the  doubly  periodic  functions  may  be  repre 
sented  as  the  quotients  of  two  Hermitean  "intermediary  functions,"  of 
which  the  Jacobi  Theta-functions  are  special  cases.  The  derivation  of 
such  functions  with  their  characteristic  properties  is  then  treated. 
Further,  by  a  method  also  due  to  Hermite,  it  is  shown  that  the  most 
general  elliptic  functions  may  be  expressed  in  terms  of  a  simple  func 
tional  element,  which  is  in  fact  the  simplest  intermediary  function. 


INTRODUCTION.  vii 

After  proving  the  theorem  that  the  most  general  elliptic  function  may 
be  expressed  algebraically  through  an  elliptic  function  of  the  second 
order  (the  simplest  kind  of  an  elliptic  function),  a  form  of  eliminant  equa 
tion  is  derived  in  which  the  derivative  appears  only  to  the  second  power. 
The  functions  connected  with  this  equation  are  treated  by  means  of  the 
Riemann  surface,  which  is  given  at  length  in  Chapter  VI,  where  also 
the  "  one-valued  functions  of  position"  are  introduced. 

The  integrals  denning  the  circular  functions  contain  radicals  under 
which  the  variable  appears  to  the  second  degree;  while  the  variable  appears 
to  the  third  or  fourth  degree  under  the  radicals  in  the  elliptic  integrals. 
It  is  therefore  natural  to  consider  the  elliptic  functions  as  the  general 
ization  of  the  circular  functions,  just  as  the  latter  functions  may  be 
regarded  as  limiting  cases  of  the  former.  The  methods  followed  by 
Legendre,  Abel  and  Jacobi  seem  the  natural  and  inevitable  methods  of 
presenting  these  functions.  History  also  gives  them  precedence.  Weier- 
strass  built  his  theory  on  the'  foundation  already  established  by  these 
earlier  mathematicians,  and  it  is  impossible  to  realize  the  real  signifi 
cance  of  Weierstrass's  functions  without  a  prior  knowledge  of  the  older 
theory.  Riemann's  theory  forms  an  important  extension  of  the  purely 
analytic  treatment  of  Legendre  and  Jacobi  as  well  as  of  the  Weierstrass- 
ian  theory.  The  characteristics  of  Riemann 's  theory  lie  on  the  one  hand 
in  the  simple  application  of  geometrical  representations  such  as  the  two- 
leaved  surface  and  its  conformal  representation  upon  the  period  paral 
lelogram,  and  on  the  other  hand  it  shows  how  the  formulas  are  founded 
synthetically  on  the  basis  of  the  fundamental  properties  of  the  functions 
and  integrals;  and  thus  a  deeper  and  a  clearer  insight  into  their  true 
nature  is  gained. 

Mr.  Poincare  has  said,  "  By  the  instrument  of  Riemann  we  see  at  a 
glance  the  general  aspects  of  things  —  like  a  traveler  who  is  examining 
from  the  peak  of  a  mountain  the  topography  of  the  plain  which  he  is 
going  to  visit  and  is  finding  his  bearings.  By  the  instrument  of  Weier- 
strass  analysis  will  in  due  course  throw  light  into  every  corner  and  make 
absolute  clearness  shine  forth." 

The  universal  laws  of  Riemann  are  particularized  in  the  one  direction 
of  the  Legendre-Jacobi  theory  and  in  the  other  direction  of  the  Weier- 
strassian  theory,  the  two  theories  being  interconnected.  Accordingly  in 
the  present  volume  the  Legendre-Jacobi  functions  are  first  developed  and 
often  side  by  side  with  them  the  corresponding  Weierstrassian  functions. 

Owing  to  a  theorem  due  to  Liouville,  we  are  able  to  show  the  real  sig 
nificance  of  the  one-valued  functions  of  position  on  the  Riemann  surface, 
viz.,  they  are  the  general  elliptic  functions.  These  one-valued  functions 
form  a  "class  of  algebraic  functions"  or  "a  closed  realm  of  rationality/' 
since  the  sum,  difference,  product,  or  quotient  of  any  two  such  functions 


viii  THEORY   OF   ELLIPTIC   FUNCTIONS. 

is  a  function  of  the  realm.  This  realm  of  rationality  is  of  the  first  order, 
corresponding  to  the  connectivity  of  the  associated  Riemann  surface,  the 
realm  of  the  ordinary  rational  functions  being  of  the  zero  order.  The 
former  realm  is  derived  from  the  latter  by  adjoining  an  algebraic  quan 
tity,  which  quantity  defines  the  Riemann  surface.  This  latter  realm, 
which  we  call  the  "  elliptic  realm,"  includes  as  special  cases  the  natural 
realm  of  all  rational  functions,  and  also  the  realm  of  the  simply  periodic 
functions.  It  therefore  follows  that  all  one-valued  analytic  functions 
which  have  algebraic  addition-theorems  form  a  closed  realm;  for  every 
element  (function)  that  belongs  to  this  elliptic  realm  has  an  algebraic 
addition-theorem.  Thus  simultaneously  with  the  development  of  the 
elliptic  functions,  the  realm  in  which  they  enter  is  shown  to  be  a  closed 
one,  and  the  reader  gradually  finds  himself  studying  these  functions  in 
their  own  realm. 

The  elliptic  or  doubly  periodic  realm  degenerates  into  a  simply  periodic 
realm  when  any  two  branch-points  coincide,  and  it  degenerates  into  the 
realm  of  rational  functions  when  any  two  pairs  of  branch-points  are  equal. 
Thus  again  it  is  seen  that  the  elliptic  realm  includes  the  three  types  of 
functions:  rational  functions,  simply  periodic  functions,  and  doubly  periodic 
functions.  In  Chapter  VII  the  eliminant  equation  is  further  simplified 
and  it  is  finally  shown  what  form  this  equation  must  have  that  the  upper 
limit  of  the  resulting  integral  be  a  one-valued  function  of  the  integral. 
The  problem  of  inversion  is  thereby  solved  in  a  remarkably  simple  manner. 
Thus  by  means  of  the  Riemann  surface,  as  it  is  possible  in  no  other  way, 
we  may  study  the  integral  as  a  one-valued  function  of  its  upper  limit  and 
vice  versa. 

In  Chapter  VIII  the  most  general  integral  involving  the  square  root  of 
an  expression  of  the  third  or  fourth  degree  in  the  variable  is  made  to 
depend  upon  three  types  of  integrals.  The  normal  forms  of  integrals  are 
derived,  and  in  particular  Weierstrass's  normal  form,  in  a  manner  which 
illustrates  the  meaning  of  the  invariants.  The  realms  of  rationality  in 
which  the  normal  forms  of  Legendre  and  of  Weierstrass  are  defined  are 
shown  to  be  equivalent. 

The  further  contents  of  this  volume  are  indicated  through  the  headings 
of  the  different  chapters.  To  be  noted  in  particular  is  Chapter  XIV,  in 
which  it  is  shown  how  the  Weierstrassian  functions  are  derived  directly 
from  those  of  Jacobi;  in  Chapter  XX  are  given  several  different  methods 
of  representing  any  doubly  periodic  function;  while  in  Chapter  XXI  we 
find  a  method  of  determining  all  analytic  functions  which  have  algebraic 
addition-theorems.  A  table  of  the  most  important  formulas  is  found  at 
the  end  of  this  volume. 

Professor  Fuchs  made  the  Riemann  surfaces  fundamental  in  his  treat 
ment  of  the  Theory  of  Functions  and  the 'Differential  Equations.  It  was 


INTRODUCTION.  ix 

my  privilege  to  hear  him  lecture  on  these  subjects,  and  the  present  work, 
so  far  as  it  has  to  do  with  the  Riemann  surfaces,  is  founded  upon  the 
theory  of  that  great  mathematician.  Although  Professor  Weierstrass 
lectured  twenty-six  times  (from  1866  to  1885)  in  the  University  of  Berlin 
on  the  theory  of  elliptic  functions  including  courses  of  lectures  on  the 
application  of  these  functions,  no  authoritative  account  of  his  work  has 
been  published,  a  quarter  of  a  century  having  in  the  meanwhile  elapsed. 
It  is  therefore  difficult  to  say  in  that  part  of  the  theory  which  bears  his 
name  what  is  due  to  him,  what  to  other  mathematicians.  I  have  derived 
considerable  help  in  this  respect  from  the  lectures  of  Professor  H.  A. 
Schwarz,  the  results  of  which  are  published  in  his  Fortneln  und  Lehrsdtze 
zum  Gebrauche  der  elliptischen  Functionen. 

While  it  has  not  been  my  purpose  to  make  the  book  encyclopedic,  I 
have  tried  to  give  the  principal  authorities  which  have  been  of  service  in  its 
preparation.  The  pedagogical  side  is  insisted  upon,  as  the  work  in  the 
form  of  lectures  is  intended  to  be  introductory  to  the  theory  in  question. 

To  Messrs.  John  Wiley  and  Sons,  Scientific  Publishers,  and  to  the 
Stanhope  Press,  I  am  under  great  obligation  for  the  courteous  co-operation 
which  has  minimized  my  labor  during  the  progress  of  printing. 

HARRIS  HANCOCK. 
2415  AUBURN  AVE., 
CINCINNATI,  OHIO, 
Nov.  1,  1909. 


CONTENTS 


CHAPTER  I 
PRELIMINARY   NOTIONS 

ARTICLE  PAGE 

1.  One-valued  function.     Regular  function.     Zeros 1 

2.  Singular  points.     Pole  or  infinity 2 

3.  Essential  singular  points 2 

4.  Remark  concerning  the  zeros  and  the  poles 3 

5.  The  point  at  infinity 4 

6.  Convergence  of  series 4 

7.  A  one-valued  function  that  is  regular  at  all  points  of  the  plane  is  a  constant  .  5 

8.  The  zeros  and  the  poles  of  a  one-valued  function  are  necessarily  isolated     .  6 

Rational  Functions 

9-10.  Methods  (1)  of  decomposing  a  rational  fraction  into  its  partial  frac 
tions;  (2)  of  representing  such  a  fraction  as  a  quotient  of  two  products 
of  linear  factors 6 

Principal  Analytical  Forms  of  Rational  Functions 

11.  First  form:    Where  the   poles  and  the  corresponding  principal   parts   are 

brought  into  evidence      8 

12.  Second  form:    Where  the  zeros  and  the  infinities  are  brought  into  evidence         9 

Trigonometric  Functions 

13.  Integral  transcendental  functions      10 

14.  Results  established  by  Cauchy 10 

15.  16.   The  fundamental  theorem  of  algebra  extended  by  Weierstrass  to  these 

integral  transcendents 12 

Infinite  Products 

17,  18.    Condition  of  convergence 14 

19.  The  infinite  products  expressed  through  infinite  series 16 

20,  21.   The  sine-function  ..:..... 17 

22.  The  cot-function ;....,.        19 

23.  Development  in  series 20 

The  General  Trigonometric  Functions 

24.  The  general  trigonometric  function  expressed  as  a  rational  function  of  the 

cot-function 22 

25.  Decomposition  into  partial  fractions 22 

26.  Expressed  as  a  quotient  of  linear  factors 25 

xi 


xii  CONTENTS. 

Analytic  Functions 

ARTICLE                                                                                                                                                               .  PAGE 

27.  Domain  of  convergence.     Analytic  continuation „ 26 

28.  Example  of  a  function  which  has  no  definite  derivative 29 

29.  The  function  is  one-valued  in  the  plane  where  the  canals  have  been  drawn  29 

30.  The  process  may  be  reversed 30 

31.  Algebraic  addition-theorems.     Definition  of  an  elliptic  function 31 

Examples 31 


CHAPTER   II 
FUNCTIONS  WHICH  HAVE  ALGEBRAIC   ADDITION-THEOREMS 

Characteristic  properties  of  such  functions  in  general.     The  one-valued  functions. 
Rational  functions  of  the  unrestricted  argument  u.     Rational  functions  of  the 

•niu 

exponential  function  e  w  . 

32.  Examples  of  functions  having  algebraic  addition-theorems  ........  33 

33.  The  addition-theorem  stated      ....................  34 

34.  Meray's  eliminant  equation     .....................  34 

35.  The  existence  of  this  equation  is  universal  for  the  functions  considered    .    .  36 

36.  A  formula  of  fundamental  importance  for  the  addition-theorems   .....  37 

37.  The  higher  derivatives  expressed  as  rational  functions  of  the  function  and 

its  first  derivative     ........................       39 

37a,  38.    Conditions  that  a  function  have  a  period    .............       41 

39.  A  form  of  the  general  integral  of  Meray's  equation   ...........        43 

The  Discussion  Restricted  to  One-valued  Functions 

40.  All  functions  which  have  the  property  that  <j>(u  +  v)  may  be  rationally  ex 

pressed  through  4>(u),  <£'(w)j  <£(v),  <}>'(v)  are  one-  valued  ........        44 

41-45.   All  rational  functions  of  the  argument  u;  and  all  rational  functions  of  the 

Urd 

exponential  function  e  w  have  algebraic  addition-theorems  and  are  such 
that 


where  F  denotes  a  rational  function     ...............       45 

46.  Example  showing  that  a  function  <!>(u)  may  be  such  that  <j>(u  +  v)  is  ration 

ally  expressible  through  <}>(u),  <j>'(u),  <j>(v),  <j>'(v)  without  having  an  alge 
braic  addition-theorem    ......................       52 

Continuation  of  the  Domain  in  which  the  Analytic  Function  <j>(u)  has  been 
Defined,  with  Proofs  that  its  Characteristic  Properties  are  Retained  in  the 
Extended  Domain 

47.  Definition  of  the  function  in  the  neighborhood  of  the  origin    .......       54 

48-50.   The  domain  of  <j>(u)  may  be  extended  to  all  finite  values  of  the  argu 

ment  u,  without  the  function  $(u)  ceasing  to  have  the  character  of  an 
integral  or  (fractional)  rational  function      .............       55 

51.  The  other  characteristic  properties  of  the  function  are  also  retained.  The 
addition-theorem,  while  limited  to  a  ring-formed  region,  exists  for  the 
whole  region  of  convergence  established  for  <£(w)  ...........  59 


CONTEXTS.  xiii 

CHAPTER   III 

THE  EXISTENCE   OF  PERIODIC  FUNCTIONS  IN  GENERAL 
Simply  Periodic  Functions.      The  Eliminant  Equation. 

ARTICLE  PAGE 

52-54.   When  the  point  at  infinity  is  an  essential  singularity,  the  function  is 

periodic 62 

55.  Functions  defined  by  their  behavior  at  infinity 67 

The  Period-Strips 

56.  The  exponential  function  takes  an  arbitrary  value  once  within  its  period- 

strip 67 

57.  The  sine-function  takes  an  arbitrary  value  twice  within  its  period-strip  .    .       69 

58.  It  is  sufficient  to  study  a  simply  periodic  function  within  the  initial  period- 

strip 70 

59.  General  form  of  a  simply  periodic  function 70 

60.  Fourier  Series 71 

61-63.   Study  of  the  simply  periodic  functions  which  are  indeterminate  for  no 

finite  value  of  the  argument;  which  are  indeterminate  at  infinity;  which 
are  one-valued,  and  which  within  a  period-strip  take  a  prescribed  value 
a  finite  number  of  times  73 

The  Eliminant  Equation 

64.  The  nature  of  the  integrals  of  this  equation 76 

65.  A  further  condition  that  an  integral  of  the  equation  be  simply  periodic. 

Unicursal  curves ' 77 

66.  A  final  condition 78 

Examples 80 


CHAPTER   IV 
DOUBLY  PERIODIC  FUNCTIONS.     THEIR  EXISTENCE.     THE  PERIODS 

67,  68.   The  existence  of  a  second  period 82 

69.  The  distance  between  two  period-points  is  finite 84 

70.  The  quotient  of  the  two  periods  cannot  be  real      .    . 85 

71.  Jacobi's  proof 86 

72.  73.   Other  proofs , 87 

74.  Existence  of  two  primitive  periods 88 

75.  The  study  of  a  doubly  periodic  function  may  be  restricted  to  a  period- 

parallelogram    89 

76.  Congruent  points 90 

77.  All  periods  may  be  expressed  through  a  pair  of  primitive  periods     ....  91 

78.  A  theorem  due  to  Jacobi 92 

79.  Pairs  of  primitive  periods  are  not  unique 93 

80.  Equivalent  pairs  of  primitive  periods.     Transformations  of  the  first  degree  .  95 

81.  Preference  given  to  certain  pairs  of  primitive  periods 96 

82.  Numerical  values                               97 


xiv  CONTENTS. 

CHAPTER  V 
CONSTRUCTION   OF  DOUBLY  PERIODIC   FUNCTIONS 

Hermite's  Intermediary  Functions.     The  Eliminant  Equation. 

ARTICLE  PAGE 

83.  An  integral  transcendental  function  which  is  doubly  periodic  is  a  constant  99 

84.  Hermite's  doubly  periodic  functions  of  the  third  sort 100 

85.  Formation  of  the  intermediary  functions 102 

86.  Condition  of  convergence 104 

87.  88.    The  Chi-function.     The  Theta-functions 106 

89.  Historical 108 

90.  Intermediary  functions  of  the  Kh  order       109 

91.  The  zeros 110 

92.  Their  number  within  a  period-parallelogram Ill 

93.  The  zero  of  the  Chi-function      113 

The  General  Doubly  Periodic  Function  Expressed  through  a  Simple  Transcendent 

94.  A  doubly  periodic  function  expressed  as  the  quotient  of  two  integral  tran 

scendental  functions 115 

95.  Expressed  through  the  Chi-function 116 

96-98.    The  Zeta-f unction.     The  doubly  periodic  function  expressed  through  the 

Zeta-function 117 

99,  100.    The  sum  of  the  residues  of  a  doubly  periodic  function  is  zero  ....  121 

101.  Liouville's  Theorem  regarding  the  infinities    .    » 122 

102.  Two  different  methods  for  the  treatment  of  doubly  periodic  functions  .    .    .  123 

The  Eliminant  Equation 

103.  The  existence  of  the  eliminant  equation  which  is  associated  with  every  one- 

valued  doubly  periodic  function 123 

104.  A  doubly  periodic  function  takes  any  value  as  often  as  it  becomes  infinite 

of  the  first  order  within  a  period-parallelogram 123 

105.  Algebraic  equation  connecting  two  doubly  periodic  functions  of  different 

orders.     Algebraic  equation  connecting  a  doubly  periodic  function  and 

its  derivative 125 

106.  The  form  of  the  eliminant  equation      126 

107.  The  form  of  the   resulting   integral.     The   inverse   sine-function.     State 

ment  of  the  "problem  of  inversion  "       126 


CHAPTER  VI 
THE  RIEMANN   SURFACE 

108.  Two-valued  functions.     Branch-points 128 

109.  The  circle  of  convergence  cannot  contain  a  branch-point 129 

110-112.    Analytic  continuation  along  two  curves  that  do  not  contain  a  branch 
point    130 

113.  The  case  where  a  circuit  is  around  a  branch-point 133 

114.  The  case  where  a  circuit  is  around  two  branch-points 133 


CONTENTS.  xv 

ARTICLE  PAGE 

115.  The  case  where  the  point  at  infinity  is  a  branch-point 134 

116.  Canals.     The  Riemann  Surf  ace  s2  =  R(z) .    .    .   .    .  134 

The  One-valued  Functions  of  Position  on  the  Riemann  Surface 

117.  Every  one- valued  function  of  position  on  the  Riemann  Surface  satisfies  a 

quadratic  equation,  whose  coefficients  are  rational  functions 137 

118.  Its  form  is  w  =  p  +  qs,  where  p  and  q  are  rational  functions  of  z    ....  138 

The  Zeros  of  the  One-valued  Functions  of  Position 

119.  The  functions  p  and  q  may  be  infinite  at  a  point  which  is  a  zero  of  w     ...  139 

120.  The  order  of  the  zero,  if  at  a  branch-point 140 

Integration 

121.  The  path  of  integration  may  lie  in  both  leaves 142 

122.  The  boundaries  of  a  portion  of  surface 143 

123.  The  residues 144 

124.  The  sum  of  the  residues  taken  over  the  complete  boundaries  of  a  portion  of 

surface 145 

125.  The  values  of  the  residues  at  branch-points 146 

126.  Application  of  Cauchy's  Theorem 148 

127.  The  one-valued  function  of  position  takes  every  value  in  the  Riemann 

Surface  an  equal  number  of  times 149 

128.  Simply  connected  surfaces . 149 

129.  130.    The  simple  case  where  there  are  two  branch-points.     The  modulus  of 

periodicity.     The  sine-function 150 

Realms  of  Rationality 

131.    Definitions.     Elements.     The  elliptic  realm 153 


CHAPTER  VII 
THE  PROBLEM   OF  INVERSION 

132.   The  problem  stated 155 

133-135.    The  eliminant  equation  further  restricted 156 

136.  The  elliptic  integral  of  the  first  kind  remains  finite  at  a  branch-point  and 

also  for  the  point  at  infinity 158 

137.  The  Riemann  Surface  in  which  the  canals  have  been  drawn 159 

138.  139.   The  moduli  of  periodicity 160 

140.  The  intermediary  functions  on  the  Riemann  Surface 162 

141.  The  quotient  of  two  such  functions  is  a  rational  function 164 

142.  The  moduli  of  periodicity  expressed  through  integrals 164 

143.  The  Riemann  Surface  having  three  finite  branch-points      165 

144-146.    The  quotient  of  the  two  moduli  of  periodicity  is  not  real 165 

147.  The  zeros  of  the  intermediary  functions 169 

148.  The  Theta-f unctions  again  introduced 171 

149.  The  sum  of  two  integrals  whose  upper  limits  are  points  one  over  the  othei  on 

the  Riemann  Surface  .  172 


xvi  CONTENTS. 

ARTICLE  PAGE 

150,  151.   The  upper  limit  expressed  as  a  quotient  of  Theta-f unctions 172 

152.  Resume 173 

153.  Remarks  of  Lejeune  Dirichlet 174 

154.  The  eliminant  equation  reduced  by  another  method 175 

155.  A  Theorem  of  Liouville 175 

156.  157.   A  Theorem  of  Briot  and  Bouquet 176 

158.  Classification  of  one-valued  functions  that  have  algebraic  addition-theorems  .      178 

159.  The  elliptic  realm  of  rationality  includes  all  one-valued  functions  which 

have  algebraic  addition-theorems 179 

CHAPTER  VIII 
ELLIPTIC  INTEGRALS  IN  GENERAL 

The  Three  Kinds  of  Integrals.     Normal  Forms. 

160-165.   The  reduction  of  the  general  integral  to  three  typical  forms.     The 

parameter      180 

Legendre's  Normal  Forms 

166-167.    Legendre's  integrals  of  the  first,  second  and  third  kinds.     The  modulus  184 

168.  The  name  "  elliptic  integral" 187 

169.  The  forms  employed  by  Weierstrass 187 

170.  Other  methods  of  deriving  Legendre's  normal  forms 188 

171.  Discussion  of  the  six  anharmonic  ratios  which  are  connected  with  the 

modulus 190 

172.  Other  methods  of  deriving  the  forms  employed  by  Weierstrass  .     ....'.  191 

173-174.   A  treatment  of  binary  forms      191 

175.   The  discriminant      193 

176-178.    The  two  fundamental  invariants  of  a  binary  form  of  the  fourth  degree  194 

179.   The  Hessian  covariant 196 

180-181.   The  two  fundamental  co variants 197 

182-183.    Hermite's  fundamental  equation  connecting  the  invariants  and  the 

covariants 198 

184.  Weierstrass's  notation 200 

185.  A   substitution   which   changes   Weierstrass's   normal   form   into   that   of 

Legendre ' 200 

186.  A  certain  absolute  invariant 201 

187.  Riemann's  normal  form 202 

188.  Further  discussion  of  the  elliptic  realm  of  rationality 202 

Examples 204 

CHAPTER  IX 

THE   MODULI   OF  PERIODICITY  FOR  THE  NORMAL  FORMS   OF 
LEGENDRE  AND   OF  WEIERSTRASS 

189.  Construction  of  the  Riemann  Surface  which  is  associated  with  the  integral 

of  Legendre's  normal  form 206 

190-192.   The  moduli  of  periodicity.     Definite  values,  in  particular  the  branch 
points,  are  taken  as  upper  limits,  and  the  values  of  the  integrals  are 

then  expressed  through  the  moduli  of  periodicity      207 

193.   The  quantities  K  and  K' 212 


CONTEXTS.  xvii 

ARTICLE  PAGE 

194-195.   The  moduli  of  periodicity  for  Weierstrass's  normal  form.     The  values 

of  the  integrals  when  branch-points  are  taken  as  the  upper  limits  .    .    .  213 

196.   The  relations  between  the  moduli  of  periodicity  for  the  normal  forms  of 

Legendre  and  of  Weierstrass 216 

197-198.   The  conformal  representations  of  the  Riemann  Surface  and  the  period- 
parallelogram    •....*.• 216 

Examples '. 219 


CHAPTER   X 
THE   JACOBI  THETA-FUNCTIONS 

199-200.  The  Theta-functions  expressed  as  infinite  series  in  terms  of  the  sine 

and  cosine .  .  > 220 

201-202.  The  Theta-functions  when  multiples  of  K  and  iK'  are  added  to  the 

argument 222 

203.  The  zeros 224 

204.  The  Theta-functions  when  the  moduli  are  interchanged 225 

Expression  of  the  Theta-Functions  in  the  Form  of  Infinite  Products 

205-206.   Products  of  trinomials  involving  the  sines  and  cosines  and  a  constant 

quantity .    *. 226 

207.  Determination  of  the  constant 228 

The  Small  Theta-Functions 

208.  Expressed  through  infinite  series .......  229 

209.  Expressed  through  infinite  products 230 

210.  Jacobi's  fundamental  theorem  for  the  addition  of  theta-functions 231 

211.  The  addition-theorems  tabulated -  .    .    .    .  234 

212.  Reason   given  for  not  expressing    the  theta-functions    through    binomial 

products    . 237 

Examples 238 


CHAPTER   XI 
THE  FUNCTIONS  sn  n,  en  a,  dn  u 

213-216.   The  elliptic  functions  expressed  through  quotients  of  the  theta-func 
tions.     Analytic  meaning  of  these  functions 239 

217.  The  zeros  of  the  elliptic  functions 244 

218.  The  argument  increased  by  quarter  and  half  periods.     The  periods  of  these 

functions 245 

219.  The  derivatives *. 246 

220.  Jacobi's  imaginary  transformation 247 

221-222.    The  co-amplitude ........... 248 

223.  Linear  transformations 248 

224.  Imaginary  argument 250 

225.  Quadratic  transformations.     Landen's  transformations 250 

226.  Development  in  powers  of  u 252 


V 

xviii  CONTENTS. 

Development  of  the  Elliptic  Functions  in  Simple  Series  of  Sines  and  Cosines 

ARTICLE  PAGE 

227.  First  method 254 

228.  Formulas  employed  by  Hermite 255 

229-231.   Second  method,  followed  by  Briot  and  Bouquet 257 

Examples 261 

CHAPTER  XII 
DOUBLY  PERIODIC  FUNCTIONS  OF  THE  SECOND   SORT 

232.  Explanation  of  the  term      264 

233.  Definitions 264 

234.  Representation  of  such  functions  in  terms  of  a  fundamental  function   .    .    .  265 

235.  Formation  of  the  fundamental  function 267 

236.  The  exceptional  case 268 

237.  Different  procedure  for  this  case 269 

238.  A  preliminary  derivation  of  the  addition-theorems  for  the  elliptic  functions  273 
239-240.    Hermite's  determination  of  the  formulas  employed  by  Jacobi  relative 

to  rotary  motion 275 

Examples 281 

CHAPTER   XIII 
ELLIPTIC  INTEGRALS   OF  THE  SECOND  KIND 

241.  Formation  of  an  integral  that  is  algebraically  infinite  at  only  one  point  .    .      282 

242.  The  addition  of  an  integral  of  the  first  kind  to  an  integral  of  the  second 

kind 284 

243.  Formation  of  an  expression  consisting  of  two  integrals  of  the  second  kind 

which  is  nowhere  infinite 285 

244.  Notation  of  Legendre  and  of  Jacobi 286 

245.  A  form  employed  by  Hermite.     The  problem  of  inversion  does  not  lead  to 

unique  results 286 

246.  The  integral  is  a  one-valued  function  of  its  argument  u 286 

247.  The    analytic   expression    of   the    integral.     Its   relation    with    the   theta- 

function 287 

248.  The  moduli  of  periodicity 289 

249.  Legendre' s  celebrated  formula 290 

250.  Jacobi's  zeta-function 291 

251.  The  properties  of  the  theta-f unction  derived  from  those  of  the  zeta-f unc 

tion;  an  insight  into  the  Weierstrassian  functions 292 

252.  The  zeta-function  expressed  in  series 295 

253.  Thomae's  notation 295 

254.  The  second  logarithmic  derivatives  are  rational  functions  of  the  upper  limit  296 
Examples  . 296 

CHAPTER   XIV 
INTRODUCTION  TO   WEIERSTRASS'S  THEORY 

255.  The  former  investigations  relative  to  the  Riemann  Surface  are  applicable 

here 298 

256.  The  transformation  of  Weierstrass's  normal  integral  into  that  of  Legendre 

gives  at  once  the  nature  and  the  periods  of  Weierstrass's  function     .    .    .     298 


CONTEXTS.  xix 

ARTICLE  PAGE 

257.  Derivation  of  the  sigma-f unction  from  the  theta-f unction 299 

258.  Definition  of  Weierstrass's  zeta-function.     The  moduli  of  periodicity   .    .    .  299 

259.  These  moduli  expressed  through  those  of  Jacobi;    relations  among    the 

moduli  of  periodicity 302 

260.  Other  sigma-functions  introduced 304 

261-262.    Sigma-functions  expressed  through  theta-f  unctions  and  Jacobi's  elliptic 

functions  expressed  through  sigma-functions 304 

263.  Jacobi's  zeta-function  expressed  through  Weierstrass's  zeta-function     .    .    .  307 
Examples , .   .    .  303 

CHAPTER    XV 

THE  WEEERSTRASSIAN  FUNCTIONS   ^n,  £n,  cru 

264.  The  Pe-function 309 

265.  The  existence  of  a  function  having  the  properties  required  of  this  function  .  311 

266.  Conditions  of  convergence 311 

267.  The  infinite  series  through  which  the  Pe-function  is  expressed,  is  absolutely 

convergent 313 

268.  The  derivative  of  the  Pe-function 314 

269.  The  periods i 316 

270.  Another  proof  that  this  function  is  doubly  periodic 316 

271.  This  function  remains  unchanged  when  a  transition  is  made  to  an  equiva 

lent  pah-  of  primitive  periods .   .   .* 317 

The  Sigma-Function 

272.  The  expression  through  which  the  sigma-f  unction  is  defined,  is  absolutely 

convergent :  expressed  as  an  infinite  product 318 

273.  Historical.     Mention  is  made  in  particular  of  the  work  of  Eisenstein    .    .    .  320 

274.  The  infinite  product  is  absolutely  convergent    . 321 

275-276.   Other  properties  of  the  sigma-f  unction 323 

The  fzz-Function 

277.  Convergence  of  the  series  through  which  this  function  is  defined 324 

278.  The  eliminant  equation  through  which  the  Pe-function  is  defined 325 

279.  The  coefficients  of  the  three  functions  defined  above  are  integral  functions 

of  the  invariants 325 

280.  Recursion   formula   for   the   coefficients  of  the   Pe-function.     The   three 

functions  expressed  as  infinite  series  in  powers  of  u      326 

281.  The  Pe-function  expressed  as  the  quotient  of  two  integral  transcendental 

functions    ....' 328 

282.  Another  expression  of  this  function 329 

283.  The  Pe-function  when  one  of  its  periods  is  infinite 332 

284-286.    The  Pe-function  expressed  through  an  infinite  series  of  exponential 

functions 332 

287-290.    The  zeta-  and  sigma-functions  expressed  through  similar  series      .    .    .  336 

291.  The   sigma-f  unction    expressed    as    an    infinite    product    of   trigonometric 

functions:  the  zeta-  and  Pe-f unctions  expressed  as  infinite  summations  of 

such  functions.     The  invariants 341 

292.  Homogeneity 343 

293.  Degeneracy * 343 

Examples 345 


xx  CONTENTS. 


CHAPTER  XVI 
THE  ADDITION-THEOREMS 

ARTICLE  PAGE 

294-295.    The  addition-theorem  for  the  theta-functions  derived  directly  from  the 

property  of  these  intermediary  functions 346 

296.  The  elliptic  functions  being  quotients  of  theta-functions  have  algebraic 

addition- theorems  which  may  be  derived  from  those  of  the  intermediary 
functions 349 

297.  Addition-theorem  for  the  integrals  of  the  second  kind 350 


Addition-Theorems  for  the  Weierstrassian  Functions 

298.  A  theorem  of  fundamental  importance  in  Weierstrass's  theory 351 

299.  Addition-theorems  for  the  sigma-functions  and  the    addition-theorem  of 

the  Pe-function  derived  therefrom  by  differentiation 352 

300-301.   Other  forms  of  the  addition-theorem  for  the  Pe-function 353 

302.  The  sigma-function  when  the  argument  is  doubled 355 

303.  Historical.     Euler  and  Lagrange 356 

304-305.    Euler's  addition-theorem  for  the  sine-function 357 

306-307.   Euler's  addition-theorem  for  the  elliptic  functions 360 

308.  The  method  of  Darboux 362 

309.  Lagrange's  direct  method  of  finding  the  algebraic  integral 365 

310.  The  algebraic  integral  in  Weierstrass's  theory  follows  directly  from  La- 

grange's  method 366 

311.  Another  derivation  of  the  addition-theorem  for  the  Pe-function     .,.-.'.  367 

312.  Another  method  of  representing  the  elliptic  functions  when  quarter  and 

half  periods  are  added  to  the  argument 367 

313.  Duplication 368 

314.  Dimidiation 368 

315-316.    Weierstrassian  functions  when  quarter  periods  are  added  to  the  argu 
ment  369 

Examples 370 


CHAPTER   XVII 
THE  SIGMA-FUNCTIONS 

317.  It  is  required  to  determine  directly  the  sigma-function  when  its  character 

istic  properties  are  assigned    .    .    .    : 372 

318.  Introduction  of  a  Fourier  Series 373 

319.  The  sigma-function  completely  determined 374 

320.  Introduction  of  the  other  sigma-functions;  their  relation  with  the  theta- 

functions    377 

321.  The  sigma-functions  expressed  through  infinite  products.     The  moduli  of 

periodicity  expressed  through  infinite  series 378 

322.  The  sigma-function  when  the  argument  is  doubled 380 

323.  The  sigma-functions  when  the  argument  is  increased  by  a  period 380 

324.  Relation  among  the  sigma-functiona 381 


CONTENTS.  xxi 

Differential  Equations  which  are  satisfied  by  Sigma-Quotients 

ARTICLE  PAGE 

325.  The  differential  equation  is  the  same  as  that  given  by  Legendre    .....  381 

326.  The  Jacobi-functions  expressed  through  products  of  sigma-functions    .    .    .  382 

327.  Other  relations  existing  among  quotients  of  sigma-functions  .    .   .    .    .   .    .  383 

328.  The  square  root  of  the  differences  of  branch-points  expressed  through  quo 

tients  of  sigma-functions     .....................  384 

329.  These  differences  uniquely  determined     ...............    .  385 

330.  The  sigma-functions  when  the  argument  is  increased  by  a  quarter-period  .  386 

331.  The  quotient  of  sigma-functions  when  the  argument  is  increased  by  a 

period     .............................  386 

332-333.    Additional   formulas   expressing   the  Jacobi-functions  through  sigma- 

functions    ............................  386 

334.  The  sigma-functions  for  equivalent  pairs  of  primitive  periods     ......  388 

Addition-Theorems  for  the  Sigma-Functions 

335.  The  addition-theorems  derived  and  tabulated  in  the  same  manner  as  has 

already  been  done  for  the  theta-functions   ..............     388 

> 

Expansion  of  the  Sigma-Functions  in  Powers  of  the  Argument 

336.  Derivation  of  the  differential  equation  which  serves  as  a  recursion-formula 

for  the  expansion  of  the  sigma-f  unction  ................     391 

Examples  .............................     394 


CHAPTER  XVIII 

THE    THETA-    AND    SIGMA-FUNCTIONS    WHEN    SPECIAL    VALUES 
ARE    GIVEN    TO    THE    ARGUMENT 

337-338.   The  theta-functions  when  the  argument  is  zero  ...........  396 

339-340.   Two  fundamental  relations  due  to  Jacobi    .............  398 

341.  The  moduli  and  the  moduli  of  periodicity  expressed  through  theta-functions  400 

342.  Other  interesting  formulas  for  the  elliptic  functions;  expressions  for  the  fourth 

roots  of  the  moduli  ........................     401 

343.  Formulas  which  arise  by  equating  different  expressions  through  which  the 

theta-functions  are  represented;  the  squares  of  theta-functions  with  zero 
arguments      ...........................      403 

344.  A  formula  due  to  Poisson    ......................      407 

345.  The  equations  connecting  the  theta-  and  sigma-functions;  relations  among 

the  Jacobi  and  the  Weierstrassian  constants    .............  408 

346.  The  Weierstrassian  moduli  of  periodicity  expressed  through  theta-functions  409 

347.  The  sigma-functions  with  quarter  periods  as  arguments      .........  410 

Examples    .............................  411 

CHAPTER  XIX 
ELLIPTIC    INTEGRALS    OF    THE    THIRD    KIND 

348.  An  integral  which  becomes  logarithmically  infinite  at  four  points  of  the  Rie- 

mann  Surface     ..........................      412 

349.  Formation  of  an  integral  which  has  only  two  logarithmic  infinities.     The 

fundamental  integral  of  the  third  kind   ...............     413 


xxii  CONTENTS. 

ARTICLE  PAGE 

350.  Three  fundamental  integrals  so  combined  as  to  make  an  integral  of  the  first 

kind 414 

351.  Construction  of  the  Riemann  Surface  upon  which  the  fundamental  integral 

is  one-valued 415 

352.  The  elementary  integral  in  Weierstrass's  normal  form 416 

353.  The  values  of  the  integrals  when  the  canals  are  crossed 417 

354-355.   The  moduli  of  periodicity 417 

356.  The  elementary  integral  of  Weierstrass  expressed  through  sigma-functions. 

Interchange  of  argument  and  parameter      419 

357.  Legendre's  normal  integral.     The  integral  of  Jacobi      420 

358.  Jacobi's  integral  expressed  through  theta-functions 420 

359.  Definite  values  given  to  the  argument 420 

360.  Another  derivation  of  the  addition-theorem  for  the  zeta-function 422 

361.  Integrals  with  imaginary  arguments 422 

362.  The  integral  expressed  through  infinite  series 423 

The  Omega-Function 

363.  Definition  of  the  Omega-function.     The  integral  of  the  third  kind  expressed 

through  this  function 423 

364.  The  Omega-function  with  imaginary  argument 424 

365.  The  Jacobi  integral  expressed  through  sigma-functions 425 

366.  Other  forms  of  integrals  of  the  third  kind 425 

Addition-Theorems  for  the  Integrals  of  the  Third  Kind 

367.  The  addition-theorem  expressed  as  the  logarithm  of  theta-functions      .    .    .  426 

368.  Other  forms  of  this  theorem 428 

369.  A  theorem  for  the  addition  of  the  parameters 428 

370.  The  addition-theorem  derived  directly  from  the  addition-theorems  of  the 

theta-functions      428 

371.  The  addition-theorem  for  Weierstrass's  integral      429 

Examples 430 

CHAPTER  XX 

METHODS    OF    REPRESENTING    ANALYTICALLY  DOUBLY    PERIODIC 
FUNCTIONS    OF    ANY    ORDER    WHICH    HAVE   EVERYWHERE   IN 
THE    FINITE    PORTION    OF    THE    PLANE    THE    CHARACTER    OF 
INTEGRAL  OR  (FRACTIONAL)  RATIONAL  FUNCTIONS 

372.  Statement  of  five  kinds  of  representations  of  such  functions 431 

373.  In  Art.  98  was  given  the  first  representation  due  to  Hermite.     This  was 

made  fundamental  throughout  this  treatise.     The  other  representations 

all  depend  upon  it 431 

374.  The  first  representation  in  the  Jacobi  theory 433 

375.  The  same  in  Weierstrass's  theory       434 

376.  The  adaptability  of  this  representation  for  integration 435 

377.  Liouville's  theorem  in  the  Weierstrassian  notation 435 

378-379.    Representation  in  the  form  of  a  quotient  of  two  products  of  theta-func 
tions  or  sigma-functions 436 

380.  A  linear  relation  among  the  zeros  and  the  infinities 438 

381.  An  application  of  the  above  representation 441 


CONTENTS.  xxiii 

ARTICLE  PAGE 

382-384.   The  fourth  manner  of  representation  in  the  form  of  a  sum  of  rational 

functions 442 

385.  The  function  expressed  as  an  infinite  product 445 

386.  Weierstrass's  proof  of  Briot  and  Bouquet's  theorem  as  stated  in  Art.  156 .  446 

387.  The  expression  of  the  general  elliptic  integral 449 

Examples 450 


CHAPTER  XXI 

THE    DETERMINATION    OF    ALL    ANALYTIC    FUNCTIONS    WHICH 
HAVE    ALGEBRAIC    ADDITION-THEOREMS 

388.  A  function  which  has  an  algebraic  addition-theorem  may  be  extended  by 

analytic  continuation  over  an  arbitrarily  large  portion  of  the  plane  without 
ceasing  to  have  the  character  of  an  algebraic  function 451 

389.  The  variable  coefficients  that  appear  in  the  expression  of  the  addition-theorem 

are  one-valued  functions      453 

390.  These  coefficients  have  algebraic  addition-theorems.     The  function  in  ques 

tion  is  the  root  of  an  algebraic  equation,  whose  coefficients  are  rationally 
expressed  through  a  one-valued  analytic  fr.r.ction,  which  function  has  an 
algebraic  addition-theorem  456 

Table  of  Formulas 458-498 


CHAPTER   I 
PRELIMINARY    NOTIONS 

ARTICLE  1.  One-valued  function.  —  A  function  of  the  complex  vari 
able  u  =  x  +  iij  is  said  to  be  one-valued  when  it  has  only  one  value  for 

each  value  of  u;  for  example,  -  ,  sin  u,  tan  u  are  one-valued  functions. 

u 

If  we  represent  the  variable  u  =  x  +  iy  by  a  point  on  the  plane  with 
coordinates  x  and  y,  we  also  speak  of  the  function  as  being  one-valued  in 
the  whole  plane,  or  in  any  part  of  the  plane  for  which  the  function  is 
denned. 

Regular  function.  —  A  one-valued  function  is  regular*  at  a  point  a 
when  we  may  develop  this  function  by  Taylor's  Theorem  within  a  circle 
with  a  as  center  in  a  convergent  series  of  the  form 


=/(a) 


the  exponents  1,  2,  .  .  .  ,  n,  .  .  .  being  positive  integers. 

The  power  series  on  the  right  is  denoted  by  P(u  —  a).  Any  such  point 
a  is  called  an  ordinary  or  regular  point  of  the  function,  and  the  function  is 
said  to  behave  regularly  f  in  the  neighborhood  of  such  a  point.  At  these 
points  the  function  has  the  character  of  an  integral  function. 

Zeros.  —  If  the  function  f(u)  is  regular  for  all  points  in  the  neighbor 
hood  of  a,  and  if  /(a)  =  0,  the  point  a  is  a  zero  of  the  function  f(u)  ;  if 
/'  (a)  ^  0,  the  point  a  is  a  simple  zero,  or  a  zero  of  the  first  order.  If  the 
derivatives  f'(a),  /"(a),  .  .  .  ,  f(n~^(a)  are  all  zero,  while  /<n)(a)  ^  0, 
the  zero  u  =  a  is  of  the  nth  order.  In  the  latter  case  the  function  f(u) 
may  be  written 

f(u)  =  (u-a)ng(u), 

*  Weierstrass,  Zur  Theorie  der  eindeutigen  analytischen  Functionen,  Werke,  Bd.  2, 
p.  77;  Berl.  Abh.  1876,  p.  11;  Abhandlungen  aus  der  Funktionenlehre,  Werke,  Bd.  2, 
p.  135;  Zur  Funktionentheorie,  Ber.  Ber.  1880,  p.  719;  Werke,  2,  p.  201. 

Mittag-Leffler,  Sur  la  representation  analytique  des  fonctions  monogines  uniformes, 
Acta  Math.,  Bd.  IV,  p.  3. 

t  "  Ich  sage  von  einer  eindeutigen  definirten  Function  einer  VeranderUchen  u,  dass  sie 
sich  in  der  Ndhe  eines  bestimmten  Werthes  UD  der  letzteren  regular  rerhalte,  wenn  sie 
sich  fur  alle  einer  gewissen  Umgebung  der  Stelle  u0  angehorigen  Werthe  von  u  in  der 
Form  einer  gewohnlichen  Potenzreihe  von  u—u^  darstellen  lasst."  Weierstrass,  Werke, 
2,  p.  295,  1883. 

1 


2  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

whe/e  g(u)  i«  a  regular  function  that  is  not  zero  for  u  =  a.     The  function 
g(u)  may  consequently  be  developed  in  a  convergent  series  of  the  form 

g(u)  =  g(a) 


ART.  2.  Singular  points.  —  If  the  one-valued  function  f(u)  is  not 
regular  at  a  definite  point  a,  we  say  that  this  point  is  a  singular  point  or 
a  singularity  of  the  function.  It  is  an  isolated  singular  point  when  we 
may  draw  around  a  as  center  a  circle  with  radius  as  small  as  we  wish, 
within  which  there  is  no  other  singularity  of  the  function. 

Pole  or  infinity.  —  A  singular  point  a  is  a  pole  or  infinity  when  it  is 
isolated  and  when  the  function  regular  in  the  vicinity  of  this  point 
becomes  at  the  point  infinite  in  the  same  way  as,  say,  the  function 


where  n  is  a  positive  integer  and  where  <f>(u)  is  a  regular  function  at  the 
point  a  and  <f>(a)  ^  0.  The  function  <j>(u)  may  be  expanded  in  a  con 
vergent  power  series  of  the  form 


so  that/(w),  when  expanded  in  the  neighborhood  of  u  =  a,  is 


(^  -  a)n 

where  F(u)  is  a  regular  function  in  the  neighborbood  of  u  =  a.     The 
constants  An,  An-i,  .  .  .  ,  AI  are  determinate,  .An  =  $(«),  etc. 
The  integer  n  is  the  order  or  degree  of  the  pole. 

The  coefficient  AI  of  •  -  is  the  residue  relative  to  the  pole  a  and 

u  —  a 

An  !  An-i  +  .    .    .  _j_       Al 

(u  —  a)n       (u  —  a)71-1  u  —  a 

is  called  the  principal  part  of  the  function  relative  to  the  pole  u  =  a. 

ART.  3.  Essential  singular  points.  —  In  the  neighborhood  of  such  a 
point,  the  function  is  completely  indeterminate.  Consider,*  for  example, 
the  function 


__ 


__ 

eu-a  U  u-a      21  (u-  a)2      3!  (u  -  a)3 

in  the  neighborhood  of  the  point  u  =  a. 

*  Cf.  Hermite,  Cours  redige  par  M.  Andoyer  (Quatrteme  edition,  1891),  p.  97. 


PRELIMINARY   NOTIONS.  3 

If  a  +  ip  be  any  arbitrary  point  whatever,  then  it  is  always  possible 
to  give  to  u  —  a  a  value  £  +  it)  as  small  as  we  wish,  such  that 

i 
6*+*'  =  a  +  1. 


For  writing  a  +  i0  =  ep  +  l'9,  the  preceding  equation  becomes 


It  follows  at  once  that 

sr\ 

and 

From  this  it  is  seen  that  £.and  y  are  completely  determined.  On  the 
other  hand  the  proposed  equation  is  satisfied  if  for  q  we  write  q  +  2  kit, 
where  k  is  an  arbitrary  integer,  since  2  IT:  is  the  period  of  the  exponential 
function.  Thus  since  q  may  be  increased  beyond  every  limit,  the  quan 

tities  £  and  T?  are  susceptible  of  becoming  as  small  as  we  wish. 

i 

The  origin  is  an  essential  singularity  of  the  function  eu.  A  character 
istic  distinction  between  the  poles  and  the  essential  singularities  is:  If 
we  take  the  inverse  of  the  proposed  function,  the  poles  are  transformed 
into  zeros;  while  an  essential  point  remains  an  essential  point,  the  recip 
rocal  of  the  function  in  the  neighborhood  of  such  a  point  being  as  the 
function  itself  completely  indeterminate.* 

In  the  present  theory  we  have  to  treat  such  functions  which  have  poles 
as  the  only  singular  points  in  the  finite  portion  of  the  plane. 

ART.  4.  Remark  concerning  the  zeros  and  the  poles.  —  If  the  point  a 
is  a  zero  of  order  n  of  the  function  f(u),  it  is  a  simple  zero  with  residue  n 

in  the  logarithmic  derivative  ^         • 

/W 
For  in  the  neighborhood  of  u  =  a  we  have 

f(u)  =  (u  -  a)n  g(u), 
where  g(a)  j^  0. 
It  follows  that 


f(u)        u  -  a       g(u) 

**         being  a  regular  function  at  the  point  u  =  a. 
9W 
Similarly  it  is  seen  that  if  u  =  a  is  a  pole  of  order  m  of  the  f  unction  /(u), 

it  is  a  simple  pole  of  residue  —  m  for  ^  ^  '  • 

*  Briot  and  Bouquet  (Fonctions  EUiptiques,  p.  94)  employ  what  seems  a  more  appro 
priate  name,  "point  d'indcter  mi  nation." 


4  THEORY   OF   ELLIPTIC   FUNCTIONS. 

For  writing 


(«-•)•' 

we  have  £$  +  -!*.+%&, 

f(u)       u  -  a       G(u) 

C"  (n^\ 

where  • — ^-~  is  a  regular  function  at  the  point  u  =  a. 
G(u) 

ART.  5.    The  point  at  infinity.  —  If  we  write  u  =  — ,  a  definite  point  in  the 

v 

w-plane   corresponds  to  a  definite  point  in  the  v-plane,  and  vice  versa. 
The  infinite  point  in  the  ^-plane  corresponds  to  the  origin  in  the  v-plane. 
Hence  if  the  function  f(u)  is  regular  at  the  point  u  =  GO  ,  the  function 

/(- )  must  be  regular  at  the  point  v  =  0.     It  must  consequently  for  small 

W 
values  of  v  in  the  vicinity  of  v  =  0  take  the  form 

=  #0  +  a\v  +  a2v2  +  •  •  •  =  P(v),    say, 

where  the  a's  are  constants.     It  follows  that  for  large  values  of  u  we  must 
have 

u       u2  un~l      un 

If  the  function  is  regular  in  the  neighborhood  of  the  point  oo ,  the  infinite 
point   is   a   zero   of   the   nth   order,  when   a0  =  0  =  ai  =  •  •   •  =  an_1; 

an  7^  0.     This   function   then  vanishes  at   infinity  as  —  (where  u  =00). 

un 

The  point  at  infinity  is  a  pole  or  an  essential  singularity  of  the  function 

f(u)y  when  v  =  0  is  a  pole  or  essential  singularity  of  /( - ).     If  u  =  oo  is 

\v/ 

a  pole,  we  must  have  for  small  values  of  v 

^2  +  .    .    .+An  +  CQ+cv  +  cv2  +  . 
V2  Vn 

where  the  A's  and  c's  are  constants;  or,  for  large  values  of  u, 


f(u)  =  Am  +  A2u2  +  -  -  -   +  Anun  +  c0  +  ^  +  ^  +  •  •  •    . 

u       Ur 

The  part  AIU  +  A2u2  +  •  •  •  +  Anun,  which  becomes  infinite  at  the 
pole  u  =  oo ,  is  the  principal  part  relative  to  this  pole  and  n  is  the  order  of  the 
pole. 

ART.  6.  Convergence  of  series. — We  have  spoken  above  of  the  con 
vergence  of  the  series  which  represents  the  function  f(u)  in  the  neighbor 
hood  of  a  point  a.  We  said  that  the  function /(w),  one-valued  in  a  defined 
region,  is  regular  at  a  point  a  of  this  region,  when  it  is  developed  by 
Taylor's  Theorem  in  a  circle  with  a  as  the  center. 


PRELIMINARY    NOTIONS.  5 

This  series  is  convergent*  within  the  circle  having  a  for  center  and  a  radius 
which  extends  to  the  nearest  singidar  point  of  the  function  f(u).  We  shall 
presuppose  the  fundamental  tests  for  absolute  convergence.  The  criterion 
for  uniform  convergence  as  stated  by  Weierstrass  is  as  follows :  The  infinite 
series  u\(z)  +  U2(z)+  u%(z)+  -  •  -  ,  the  individual  terms  of  which  are 
functions  of  z  defined  for  a  fixed  interval,  converges  uniformly  within  this 
interval,  provided  there  exists  an  absolutely  convergent  series, 

Ml  +  J/2  +  -  •  '  , 

where  the  M's  are  quantities  independent  of  z  and  are  such-  that  within  the 
fixed  interval  the  following  inequality  is  true: 

j  iin(z)  ,  =  Mn,    where  n  =   «. 

/JL  being  a  fixed  integer.  (See  Osgood,  Lehrbuch  der  Funktionentheorie, 
p.  75.) 

ART.  7.  A  one-valued  function  that  is  regular  at  all  points  of  the  plane 
(finite  and  infinite}  is  a  constant. 

For  the  function  supposed  regular  at  u  =  0  is  developable  in  the  series 

f(u)  =  a0  4-  a\u  +  a2u2  4-  •  •   •  =  P(u),    say, 

which  is  convergent  within  a  circle  which  may  extend  to  infinity,  since  by 
hypothesis  there  are  no  singular  points  in  the  plane. 

Writing  u  =  - ,  the  expansion  in  the  neighborhood  of  infinity  is 


This  function  being  by  hypothesis  regular  in  the  neighborhood  of  infinity, 
can  contain  no  negative  powers. 

It  follows  that  a  i  =  0  =  a2  =  a3  =   .  .  .  ,  and  consequently 

/(")  - 

Another  statement  of  this  theorem  is  the  following:  A  one-valued  function 
that  is  finite  at  all  points  of  the  plane  (including  the  infinite  point)  is  a  constant. 

For  at  each  one  of  its  poles  a  one-valued  function  becomes  infinite.  It 
may  also  be  shown  that  if  the  variable  u  tends  towards  an  essential  singu 
larity  in  a  manner  which  has  been  suitably  chosen,  the  modulus  of  the 
function  increases  beyond  limit.  If  then  a  one-valued  function  is  every- 

*  See  Cauchy,  Cours  d' Analyse  de  I'Ecole  Royale  Poly  technique,  l^re  Partie.  Analyse 
Algcbrique,  Chapitre  9,  §  2,  Theoreme  I,  p.  286.  Paris.  1821.  Unless  stated  other 
wise,  by  "convergent"  is  meant  absolutely  convergent.  (See  Osgood,  Lehrbuch  der 
Funktionentheorie,  pp.  75  et  seq.;  pp.  285  et  seq.};  and  when  the  variable  enters,  uni 
formly  convergent.  In  the  latter  case  by  "within  the  circle  of  convergence"  we 
understand  "within  any  interval  that  lies  wholly  within  this  circle." 


6  THEORY   OF   ELLIPTIC   FUNCTIONS. 

where  finite,  it  cannot  have  singular  points;  it  is  regular  throughout  the 
whole  plane  and  reduces  to  a  constant. 

ART.  8.  The  zeros  and  the  poles  of  a  one-valued  function,  which  has  no 
other  singularities  than  poles  in  the  finite  portion  of  the  plane,  are  neces 
sarily  isolated  the  one  from  the  other. 

By  this  we  mean  to  say  that  there  cannot  exist  a  point  a  of  the  plane  in 
whose  immediate  neighborhood  there  are  an  infinite  number  of  poles  or  an 
infinite  number  of  zeros.  In  other  words,  wherever  the  point  a  is  situated, 
one  may  always  draw  around  a  as  center  a  circle  with  radius  sufficiently 
small  that  within  the  circle  there  are  (1)  neither  zero  nor  pole;  or  (2)  a  zero 
but  no  pole;  or  (3)  a  pole  but  no  zero. 

This  follows  immediately  from  the  preceding  developments.  For  if  a 
point  a  is  taken  in  the  plane,  three  cases  are  possible:  (1)  the  function  f(u) 
may  be  regular  at  a  without  vanishing  at  this  point;  or  (2)  the  point  a  is 
a  zero  of  f(u) ;  or  (3)  the  point  a  is  a  pole  of  f(u).  In  the  first  case  we  may 
draw  about  a  as  center  a  circle  with  radius  sufficiently  small  that  within 
the  circle  there  is  neither  zero  nor  pole;  in  the  second  case  we  may  draw  a 
circle  sufficiently  small  that  it  does  not  contain  a  pole  and  contains  the 
only  zero  u  =  a,  and  similarly  in  the  third  case. 

It  follows  that  if  for  a  one-valued  function  there  exists  a  point  a  such  that 
within  an  area  as  small  as  we  choose  inclosing  this  point  there  exists  an 
infinity  of  poles  or  an  infinity  of  zeros,  this  point  is  an  essential  singularity. 
The  function  is  not  regular  at  this  point.  As  examples  of  what  has  been 
said  are  the  rational  functions  and  the  trigonometric  functions, which  shall 
be  first  studied  as  introductory  to  the  general  theory  of  elliptic  functions. 

RATIONAL   FUNCTIONS. 

ART.  9.  Methods  are  given  here,  (1)  of  decomposing  a  rational  fraction 
into  its  simple  (or  partial)  fractions;  (2)  of  representing  such  a  fraction  as 
a  quotient  of  two  products  of  linear  factors.  The  same  methods  will  be 
adopted  later  in  the  general  theory  of  elliptic  functions,  there  existing 
analogous  relations  for  these  functions. 

Consider  first  as  a  particular  case  *  the  function 

/(«)  = 


(u  -  1)  (u  -  2) 


\  /          \  -  / 

which  is  regular  at  all  finite  points  of  the  plane  except  the  points  u  =  1 
and  u  =  2.     These  points  are  poles  of  the  first  order.     The  principal  part 

= 


. 

of  f(u)  relative  to  the  pole  u  =  I  is 


say, 


*  See  Appell  et  Lacour,  Fonctions  Elliptiques,  p.  7. 


PRELIMINARY   NOTIONS.  1 

as  is  seen  by  noting  that  the  difference 

f(u)  -  0i  (M) 

is  regular  at  the  point  u  =  1.     The  residue  relative  to  the  pole  u  =  I  is  —  1. 
Similarly  the  principal  part  relative  to  the  pole  u  =  2  is 

*»(«)  =  —^-5, 

i£  —  2 
with  the  residue  2. 

At  the  point  u  =  «D  the  function  is  regular,  for 


(1  -  v)  (1  -  2  v) 

is  a  regular  function  at  the  point  v  =  0. 

It  is  further  seen  that  v  =  0  or  u  =  oo  is  a  simple  zero.  The  function 
f(u)  has  then  two  simple  poles  u  =  1,  u  =  2  and  two  simple  zeros  u  =  0, 
u  =  QC  .  The  function  is  said  to  be  of  order  or  degree  2. 

It  may  also  be  observed  that  the  equation 

/(«)  =  c 

has  two  roots,  whatever  be  the  constant  C.  Further,  since  the  functions 
0i(u)  and  02  (u)  are  everywhere  regular  except  at  the  poles  u  =  1,  u  =  2, 
the  difference 

/OO  -  0i 00  -  0200 

is  a  function  that  is  everywhere  regular.     It  is  therefore  a  constant,  and 
since  /(M),  0iOO>  0200  all  vanish  for  M  =  oc ,  this  constant  is  zero. 
We  therefore  have 

/OO  =  0iOO  +  0200, 

a  formula,  which  gives  immediately  the  decomposition  of  the  rational 
function  f(u)  into  its  simple  fractions. 

ART.   10.     The  general  case. — A  rational  function 

/OO  ' 


1  +  •  •  •  +  bn        Q(u) 

where  Qi  and  Q  are  integral  functions  (polynomials)  of  degree  m  and  n, 
is  a  function  which  has  no  other  singularities  than  poles  in  the  finite  portion 
of  the  plane  or  at  infinity.  At  a  finite  distance  it  has  as  poles  the  roots  of 
QOO  =  0-  The  number  of  these  poles  at  a  finite  distance,  where  each  is 
counted  with  its  order  of  multiplicity,  is  n. 

1°.  If  m  >  n,  the  point  at  oo  is  a  pole  of  order  m  —  n.  Hence  the 
total  number  of  poles  at  finite  and  infinite  distances  is  n  4-  m  —  n  =  m. 

There  are  also  m  zeros,  viz.,  the  roots  of  Q\(u)  =  0.  It  is  thus  seen 
that  the  function  f(u)  has  m  zeros  and  m  poles.  We  say  that  it  is  of 


8  THEORY    OF   ELLIPTIC   FUNCTIONS. 

order  or  degree  m.  The  equation/(w)  =  C  has  m  roots,  whatever  the  value 
of  the  constant  C. 

2°.  If  n  >  m,  the  point  oo  is  a  zero  of  order  n  —  m.  The  function  has 
n  poles  and  an  equal  number  of  zeros.  For  there  are  m  zeros  at  finite 
distances,  viz.,  the  roots  of  Q\(u)  =  0  and  n  —  m  zeros  at  infinity.  The 
function  is  of  order  n  and  the  equation /(ti)  =  C  has  n  roots. 

3°.  If  m  =  n,  the  point  at  infinity  is  neither  a  pole  nor  a  zero.  There 
are  also  here  as  many  zeros  as  infinities,  and  the  function  is  of  order  m  =  n. 

It  follows  that  a  rational  function  f(u)  has  always  in  the  whole  plane, 
including  infinity,  as  many  zeros  as  poles.  The  number  of  zeros  or  poles 
is  the  order  of  the  function,  and  the  equation  f(u)  =  C,  where  C  is  an 
arbitrary  constant,  has  a  number  of  roots  equal  to  the  order  of  the  func 
tion  f(u).  In  particular  we  note  that  the  rational  functions  have  only  polar 
singularities. 

PRINCIPAL  ANALYTICAL  FORMS  OF  RATIONAL  FUNCTIONS. 

ART.  11.  First  form:  where  the  poles  and  the  corresponding  principal 
parts  are  brought  into  evidence.  Decomposition  into  simple  fractions. 

Let  01,  a2,  .  .  .  ,  av  be  poles  of  order  ni,  n2,  .  .  .  ,  nv  of  the  function 
f(u)  and  let  the  principal  parts  with  respect  to  these  poles  be 


u  —  av       (u  —  av)2  (u  —  av}nv 

Further  for  the  most  general  case,  suppose  that  the  point  oo  is  also  a 
pole,  which  is  the  case  in  the  previous  Article  when  m  >  n;  and  let  the 
principal  part  relative  to  this  pole  be 

(f)(u)  =  AIQU  +  A2ou2  +••••+  Asous, 

where  s  =  m  —  n  is  the  order  of  the  pole. 

Since  each  of  the  principal  parts  is  everywhere  regular  except  at  the 
associated  pole,  the  difference 

f(u)  —  <£i  O)  —  02  (w)  —  •  •  •  -  0(w) 

is  regular  everywhere  including  infinity  and  consequently  is  a  constant, 
=  A,  say. 


PKELIMINARY   NOTIONS. 

It  follows  that 

f(u)  =  A  +  Alou  +  A2ou2  +  •  •  •  + 


ffi  \«  -  ai       (u  ~  °f)2  (u  ~  a>i] 

where  the  index  i  refers  to  the  indices  of  the  poles  ai,  a2,  .  .  .  ,  av.     This 

formula  may  be  written  in  a  somewhat  simpler  form  if  we  symbolize  -  by 

u 

u  —  a0,  where  a0  =  »,  and  let  no  =  s.     We  then  have 


f(u)  =A 


where    the    summation    index    i   refers    to    the    indices    of    the    poles 
ai,  a2,  .  .  .  ,  a*,  a0. 

If  we  put  -  =  v,-,  we  have  finallv 
u  —  ai 


-  A 


The  formula  is  convenient  especially  for  the  integration  of  a   rational 
function.* 

ART.  12.  Second  form:  where  the  zeros  and  the  infinities  are  brought  into 
evidence.  It  is  sufficient  here  to  decompose  the  polynomials  Qi(u)  and 
Q(u)  of  the  preceding  article  into  their  linear  factors,  so  that 

f(u)  =  C  (^  ~  ci)  (M  -  c2)  .  .  .   (M  -  cm) 
(u  -  61)  (M  -  62)  ...   (M  -  6,)' 

where  C  is  a  constant.     Of  course,  some  of  the  factors  may  be  equal.     We 
may  derive  the  second  form  from  the  first  by  noting  (Art.  4)  that 


. 

f(u)        u  —  GI       u  —  c2 

1  1 


u  —  bi       u  —  b2  u  —  bn 

Integrating  and  passing  from  logarithms  to  numbers,  we  have  the  form 
required. 

In  the  next  Chapter  it  will  be  shown  that  any  rational  function  f(u)  has 
an  algebraic  addition-theorem;  that  is,  if  u  and  v  are  two  independent 
variables,  f(u  +  v)  may  be  expressed  algebraically  in  terms  of  f(u)  and/(v). 

*  Cf.  Appell  et  Lacour,  loc.  cit.,  p.  9. 


10  THEORY   OF   ELLIPTIC    FUNCTIONS 


TRIGONOMETRIC   FUNCTIONS. 

ART.  13.  In  the  presentation  of  some  of  the  fundamental  properties 
of  the  trigonometric  functions  we  shall  apply  methods  which  are  later 
used  in  a  similar  manner  in  the  theory  of  the  elliptic  functions. 

The  polynomial  a0  +  a\u  +  a2u2  +  •  •  •  +  anun  =  F(u)  is  a  one- 
valued  function  with  a  finite  number  of  terms  each  having  a  positive  inte 
gral  exponent.  This  integral  function  is  of  the  nth  degree. 

Another  class  of  one-valued  functions  are  those  where  n  has  an  infinite 
value.  Such  functions,  when  convergent  for  all  finite  values  of  the  vari 
able,  are  known  as  integral  transcendental  functions. 

For  example, 


is  a  series  which  is  convergent  for  all  finite  values  of  u  and  is  a  regular 
function  at  all  points  at  a  finite  distance  from  the  origin.  It  becomes  zero 
for  the  values 

U    =    0,    ±   71,    ±    2  7T,    ±    3  7T,     •     -     •        . 

We  know  that  the  decomposition  of  a  polynomial  into  a  product  of 
linear  factors  is  the  fundamental  problem  of  algebra.  It  is  natural  to 
seek  whether  the  integral  transcendents  may  not  also  be  decomposed  into 
their  prime  factors.  Euler  gave  the  celebrated  formula 

sin  nu 

7tU 

a  formula  which  is  true  for  every  finite  value  of  u.  Cauchy  was  the  first 
to  treat  the  subject  in  general.  Although  he  did  not  complete  the  theory, 
he  recognized  that  if  a  is  a  root  of  the  integral  transcendent  f(u),  it  is  neces 
sary  in  many  cases  to  join  to  the  product  of  the  infinite  number  of  factors 

such  as  1  —  -  a  certain  exponential  factor  ep:u\  where  P(u)  is  a  power 
a 

series  in  positive  powers  of  u.  Weierstrass  gave  a  complete  treatment  of 
this  subject. 

ART.  14.  We  may  establish  first  the  results  derived  by  Cauchy.  With 
Hermite  (loc.  cit.,  p.  84)  suppose  that  a\,  a%,  a3,  .  .  .  are  the  roots  of  the 
integral  transcendental  function  f(u)  which  are  arranged  in  the  order  of 
increasing  moduli.  Further  suppose  that  they  are  all  different  and  none 
of  them  is  zero. 

Suppose  first  that  the  series 


mod  di 


PRELIMINARY   NOTIONS.  11 

formed  by  the  inverse  of  the  moduli  of  the  roots,  is  convergent.     The 
same  will  (as  shown  below)  also  be  true  of  the  series 


mod  (a;  —  u) 
whatever  the  value  of  u,  excepting  the  values  u  =  (LI,  a2,  .  .  .  ,  which 


l=X> 


i 

make  the  series  infinite.     It  follows  then  that  V  -  will  represent  an 

fl  u  -  at 

analytic  function  in  the  whole  plane.* 

To  prove  the  above  statement  consider  the  two  infinite  series  2un  and 
Stfn,  of  which  the  first  is  convergent. 

The  second  series  will  also  be  convergent  if  we  have 

vn  <  kun     (k  constant) 


for  all  values  of  n  starting  with  a  certain  limit.     If  we  write  Un  = 

i 

,  the  condition  just  written  is 


mod  (an  —  u) 


mod  an        <  j. 


mod  (an  —  u) 
From  the  inequality 

mod  an  <  mod  (an  —  u)  +  mod  u, 
we  have 

mod  On  ,    ,          mod  u 


mod  (an  —  u)  mod  (an  —  u) 

which  demonstrates  the  theorem  since — — decreases  indefinitely 

mod  (an  —  u) 

when  n  increases. 

It  is  seen  at  once  that 


,  ,  ,  u  —  an 

is  a  regular  function  for  all  finite  points  of  the  plane.     This  difference  we 

may  represent  by  G'(u)  =  — **£• 

du 

We  thus  have 


Multiplying  by  du  and  integrating,  taking  zero  as  the  lower  limit,  we 
have 


iu,   say; 


*  See  Osgood,  Lehrbuch  der  Funktionentheorie,  p.  75  and  p.  259. 


12  THEORY    OF   ELLIPTIC    FUNCTIONS. 

where  the  product  is  to  be  taken  over  the  finite  or  infinite  number  of  factors 


a  i  0,2 

This  result  is  due  to  Cauchy,  Exercises  de  Mathematiques,  IV. 

ART.  15.  We  may  next  consider  the  general  case  and,  following  the 
methods  of  Mr.  Mittag-Le  frier,*  establish  the  important  results  of  Weier- 
strass  |  who  extended  to  these  integral  transcendents  the  fundamental 

theorem  of  algebra.     When  the  series  of  the  preceding  article  V  -  -  — 

,  *•*  mod  an 

is  not  convergent,  the  sum  "V  -  no   longer  represents  an  analytic 

^  u  -  an 

function;  but  by  subtracting  from  each  term  a  part  of  its  development 
arranged  according  to  decreasing  powers  of  n,  Mr.  Mittag-Leffler  has  shown 
that  it  is  possible  to  form  with  these  differences  an  absolutely  convergent 
series. 

Let  Pu(u)  -  1  +  \  +  •  •  •  +  u—> 

an      an2  an» 


so  that 


P.(M) 


, 

u  —  an  ana>(u  —  On) 


We  may  next  show  that  by  a  suitable  choice  of  CD  we   may  render  the 
series 

—  +  *>.(«),    or 


an"(u  -  On) 
convergent. 


In  the  first  place  it  may  happen  that        --  being  divergent,  the 

*~*  mod  an 

series  formed  by  raising  each  term  of  the  divergent  series  to  a  certain 
power  is  convergent.     For  example,  in  the  case  of  the  divergent  har 

monic  series  V  -  ,  we  know  that  V  —  ,  where  u.  >  1  ,  is  convergent. 
^4  n  ^  n*  , 

Hence  we  may  fix  a  number  a>  such  that  the  series    > 

^ 

convergent. 

We  may  then  conclude  from  this  series  the  convergence  of 


s 
mod 


V  -  -  -  -  -  -  ,   and  consequently  of    V  —  —  -  -  -  • 

^  mod  an"(u  -  an)  ^  an"(u  -  an) 

*  See  Mittag-Leffler,  loc.  cit.,  p.  38;  and  Comptes  rendus,  t.  94,  pp.  414,  511,  713, 
781,  938,  1040,  1105,  1163;  t.  95,  p.  335. 

t  Weierstrass,  Werke,  Bd.  Ill,  p.  100.  See  also  Casorati,  Aggiunte  a  recenti  lavori 
dei  Sigi  Weierstrass  e  Mittag-Leffler;  Annali  di  Matematica,  serie  ii,  t.  X;  Harkness 
and  Morley,  Theory  of  Functions,  p.  188;  Forsyth,  Theory  of  Functions,  p.  335. 


PKELIMINARY   NOTIONS.  13 

For,  if  we  put 

t  vn 


mod  anal+1  mod  anw(an  —  u) 

we  have  for  the  ratio  —  the  same  value  as  before, 
Un 

vn  mod  an 


un      mod  (an  —  u) 

We  must,  however,  always  know  that  we  are  passing  to  a  convergent 
series  when  \ve  raise  each  term  of  the  divergent  series  to  a  certain  power. 

For  example,*  consider  the  divergent  series  V .     It  is  seen  that 

x  ^  log  n 

the  series  V is  also  divergent,  however  great  aj  be  taken. 

~*  (log  n) 

For  writing 


(log  2)-       (log  3)-  (log  n)' 

it  is  seen  that 

S    >   n  ~  l  - 
(log  n)a 

Note  that 

n  -  1  n  1 


(log  n)w       (logn)u       (logn)"' 

and  that  the  first  term  on  the  right  increases  with  increasing  n,  while 
the  second  term  tends  towards  zero.     The  series  is  therefore  divergent. 

ART.  16.  In  such  cases  as  the  above  Weierstrass  took  for  co  a  value 
which  changes  with  n.  With  W^eierstrass  write  a>  =  n  —  1.  The  given 
series  may  be  written 


This  series  is  convergent;  for  writing 
Un  =  mod 


f1- 

\  On 


it  is  seen  that  vUn  tends  towards  zero  for  n  =  oo .  We  know  (cf.  Art.  86) 
that  it  is  sufficient  for  this  limit  to  be  less  than  unity  for  a  convergent 
series. 

It  follows  as  before  that  the  expression 

/'(«) 


*  This  example  is  due  to  Mr.  Stern  and  cited  by  Hermite,  loc.  cit.,  p.  86. 


14  THEORY   OF   ELLIPTIC   FUNCTIONS. 

is  a  function  that  remains  regular  for  all  finite  values  of  u.  It  must 
therefore  be  expressible  in  a  convergent  power  series  in  ascending  powers 
of  u. 

Write  this  series.  =  — ^Hl  •  and  for  brevity  write 
du 


so  that 


ro  Q>n       £  Q<n  wanw  \an 

We  have  at  once 


which  formula  gives  an  analytic  expression,  in  which  the  roots  are  set 
forth,  of  the  integral  transcendental  function. 

The    quantities    (1  --  \e  w\a»)    are    called    primary   functions*    by 
\         On/ 

Weierstrass. 

Suppose  next  that  f(u)  has  equal  roots,  say,  of  the  pih  order  of  multi 
plicity.  We  see  immediately  that  the  formula  does  not  undergo  any 
analytic  modification,  it  being  sufficient  to  raise  the  corresponding 
primary  factor  to  the  pth  power. 

Finally  if  we  admit  the  case  of  a  function  having  a  zero  root  of  the  qth 

order,  we  have  only  to  proceed  with  the  quotient  £^2*1,  the  result  differing 

from  the  preceding  only  by  the  presence  of  the  factor  UQ.     (See  Hermite, 
loc.  cit.) 

INFINITE  PRODUCTS. 

ART.   17.     It  may  be  shown  that  the  infinite  product 
(1  +  ax)  (1  +  a2)  .  .  .   (1  +  a»)  .  .  . 
has  a  definite  value,  if 


represents  a  converge  at  series.! 

*  See  Osgood,  Ency.  der  math.  Wiss.,  Band  II2,  Heft  1,  pp.  78  et  seq.;  Forsyth, 
Theory  of  Functions,  pp.  92  et  seq.;  Weierstrass,  Werke,  II,  p.  100;  Harkness  and  Morley, 
Theory  of  Functions,  p.  190. 

f  Cf.  Mittag-Leffler,  Acta  Math.,  Vol.  IV,  pp.  SQetseq.;  Dini,  Ann.  di  mat.  (2),  2,  1870, 
p.  35;  Harkness  and  Morley,  Theory  of  Functions,  p.  82;  and  especially  Pringsheim, 
Ueber  die  Convergenz  unendlicher  Producte,  Math.  Ann.,  Bd.  33;  Weierstrass,  Werke, 
I,  p.  173. 


PRELIMINARY   NOTIONS.  15 

For  write 

Pn  =  (1  +  ai)  (1  +  a2)  .  .  .   (1  +  an). 

Then  evidently 

Pn   —  Pn  -  1   =  #nPn  -  1  , 

and 

Pn  =  1  +  ai  +  a2Pi  +  a3P2  +  •  '  •  •  +  cinPn-i. 

Hence  when  n  becomes  indefinitely  large,  the  series  Pn  will  tend  towards 
a  definite  limit  if  the  series 


1  +ai  +a2Pl  +  a3P2  +  •   •   •  +  anPn-1  +an  +  1Pn+  •   •  •          (1) 

is  convergent,  the  limit,  if  there  is  one,  being  the  sum  of  this  series. 

Consider  first  the  case  where  the  quantities  a\t  a2j  .  .  .  are  real  and 
positive  or  zero.  The  quantities  PI,  P2,  .  .  .  are  then  at  least  equal  to 
unity,  and  consequently,  in  order  that  the  series  (1)  be  convergent,  it  is 
necessary  that  the  series 

ai  +  a2  +  a3  +  •  •  •  +  an  +  -  •  •  (2) 

be  convergent. 

Further,  if  (2)  is  convergent,  it  may  be  shown  as  follows  that  (1)  is 
convergent. 

The  product 

Pn  =  (1  +  ai)  (1  +  a2)  .  .  .   (1  +  an), 

when  developed  is 

1  +  01  +  a2  -h  •  •  •  +  an  +  aia2  +  •   •   •  +  a^2  •   •    an. 

Writing 

An  =  ai  +  a2  +  •  •  •  +  an     and 

A  =  al  +  a2  -f-  •   •  •  +  an  +  an  +  i  +  •  •  •  , 
it  is  seen  that 

A  A    2  An 

P      S    1    _l_  ^n  _l_  •"•"_  J_  _1_  ^n 

IT     »  '^T' 

or  Pn  <  eAn  <  eA, 

which  proves  the  proposition. 

Next  let  the  quantities  «i,  a2,  .  .  .  ,  an,  .  .  .  ,  previously  supposed  to 
be  real  and  positive,  take  any  values. 

Then  the  series 

1  +  ai  +  a2  -f  •   •   •  +  an  +  a,ia2  +  •  •   -  +  a\a^  •  •  an 

is  evidently  convergent  if  the  corresponding  series  made  by  taking  the 
absolute  values  of  the  a's  is  convergent. 

n  =  x 

Hence  the  condition  for  the  convergence  of  the  product  JJ  (1  +  an)  is 

n=x  n=l 

that  the  series  'V  |  an  \  be  convergent. 


16  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  18.     If  further  the  series 

|  al  |  +  |  a2     +     a3  |  +  •  •  -  +    an  |  +  •  •  •  (3) 

is  convergent,  the  product 

(1  +  aiu)  (1  +  a2u)  (1  +  a3u)  .  .  .   (1  +  anu)  ...  (4) 

is  convergent  for  all  values  of  the  variable  u,  except  infinity.     For  if  r  is 
the  modulus  of  u,  the  series 

|  ai    r  +  |  a2    r  +  •  •  •  +  |  an  \  r  +  •   •   • 

is  convergent  whatever  be  the  modulus  r. 

ART.  19.  We  shall  next  show  that  when  the  series  (3)  is  convergent,  the 
product  (4)  may  be  expressed  as  an  integral  power  series  P(u)  which  is  con 
vergent  *  for  all  finite  values  of  u. 

Consider  the  product  of  n  real  factors 

Pn(r)  =  (1  +  air)  (1  +  a2r)  .  .  .   (1  +  anr), 

ai,  a2,  .  .  .  being  taken  as  real  quantities  and  positive.     Let  these  n 
factors  be  multiplied  together. 

If  si(n)  denotes  the  sum  of  the  n  quantities  ab  a2,  .  .  .  an;  s2(n)  the  sum 
of  the  products  of  these  n  quantities  taken  two  at  a  time;  s3(n)  the  sum 
of  the  products  taken  three  at  a  time,  etc.,  we  will  have 

pn(r)  =  1  +  Sl(«)r  +  s2(n)r2  +  •   •   •  +  sm(n)rm  +  •   •   •  +  sn(n)rn. 


Since  any  term  sm(n)rm  is  less  than  Pn(r)  or  its  limit  P(r),  where  n  =  oo, 
it  follows  that  sw(n)rm  tends  toward  a  definite  limit  smrm  when  n  increases 
indefinitely;  thus  the  sum  sm(n)  of  the  products  taken  m  at  a  time  of  the  n 
first  terms  of  the  convergent  series 

ai  +  a2  H-  a3  +  •  •  • 

tends  toward  a  definite  limit  sm  when  n  increases  indefinitely. 
But  since 

Pn(r)  >  1  +  Si<nV  +  s2(n)r2  +  .   .   .  +  sm(wVm, 

if  leaving  m  fixed  we  let  n  increase  indefinitely,  it  is  seen  that 
P(r)  >  1  +  sir  +  s2r2  +  .   .   .  +  smrm. 

Since  the  sum  Sm(r)  of  the  m  first  terms  (m  indefinitely  large)  of  the  series 
1  +  sir  +  s2r2  +.••••  (5) 

is  less  than  a  finite  quantity  P(r),  we  conclude  that  this  sum  tends  toward 
a  limit  S(r)  which  is  less  than  or  equal  to  P(r). 

*  See  Briot  et  Bouquet,  Fonctions  Elliptiques,  pp.  301  et  seq.;  Osgood,  Lehrbuch 
der  Funktionentheorie,  pp.  460  et  seq.;  Tannery  et  Molk,  Fonctions  Elliptiques,  t.  I, 
pp.  28  et  seq.;  Picard,  Traite  d'  Analyse,  I,  2,  p.  136;  Bromwich,  Theory  of  Infinite 
Series,  pp.  101  et  seq. 


PRELIMINARY   NOTIONS.  17 

On  the  other  hand,  each  of  the  terms  of  the  product 

Pn(r)  =  1  +  s^nh  +  s2(n)r2  +  •   •   •  +  sn(n)rn 
being  less  than  the  corresponding  term  of  the  sum 

Sn(r)  =  1  +  sir  +  s2r2  +  •   •   •  +  snrn, 

the  sum  /Sn(r)  is  greater  than  Pn(r)  and   consequently  its  limit  S(r)  is 
greater  than  or  equal  to  P(r). 

It  follows  that  S(r)  =  P(r). 

Consider  next  the  product  of  n  imaginary  factors 

Pn(u)    =    (1    +   aitt)    (1    +   CL2U)    .    .    .     (1    +   dnu), 


where  |  a\  |  +  |  a2  |  +  |  a3  |  +  •   •   •  is  a  convergent  series. 
It  follows  as  above  that 


Pn(U)    =    1    +    ff^U  +    <72(n)W2   +   •     •    •   +    <7m(n}Um  +   -    •     •   +  <7n(n)l*n. 

Any  coefficient  am(n)  is  a  sum  of  imaginary  terms  whose  moduli  form 
quantities  corresponding  to  sm(n)  above.  Consequently  when  n  increases 
indefinitely,  since  sm(n)  tends  towards  a  limit  sm,  the  sum  am^  tends 
towards  a  limit  om. 

The  series 

I  +  oiu  +  o2u2  +  <73u3  +.'.-.-  P(u),     say,  (6) 


is  convergent,  since  the  moduli  of  its  terms  are  less  than  the  correspond 
ing  terms  of  (5). 

The  sum  Sn(u)  of  the  n  first  terms  of  this  series  contains  all  the  terms  of 
Pn(u).  Further,  the  terms  of  the  difference  Sn(u)—  Pn(u)  have  for  their 
moduli  the  corresponding  terms  of  the  difference  <Sn(r)  —  Pn(r)  and  con 
sequently  tend  towards  zero,  when  n  increases  indefinitely.  We  conclude 
that  Sn(u)  tends  towards  a  limit  P(u). 

Thus  the  function  defined  by  the  product  (4)  is  developable  in  a  uni 
formly  convergent  series  (6)  arranged  according  to  increasing  powers  of  u. 

ART.  20.     The  sine-function.  —  As  an  example  of  Art.  16,  we  note  that 


the  function  f(u)  =  sm  ~u  has  for  its  roots  all  the  positive  and  negative 
nu 

integers  ±  1,  ±  2,  ±  3,  •  •  •  . 

The   series  V  -  is  here  divergent,  but  the  series  V  -  -  -  —  —  is 
**  *  2 


convergent. 

We  may  consequently  put  a>  =  1  in  Weierstrass's  formula.     The  primary 
factors  are  therefore 

u 

1  -  -}e*. 


18  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Noting  that  f(o)  =  1,   and   admitting*  that  G(u)  =  0  (see  Vivanti- 
Gutzmer,  Eindeutige  analytische  Funktionen,  p.  163),  we  have  the  formula 

sin  Tin 
nu 

n  =  ±  1,  ±  2,  ±  3,  -  -  -     . 

Uniting  the  integers  that  are  equal  and  of  opposite  sign  we  have  Euler's 
formula : 

sin  nu 


The  periodic  property  of  the  sine-function  may  be  deduced  from  this 
definition.     For  write 

F(u)  =  Au(u  —  1)  (u  —  2)  .  .  .   (u  —  n)     multiplied  by 
(u  +  1)  (u  4-  2)  .  .  .   (u  4-  n\ 

where  A  is  a  constant. 

Changing  u  into  u  +  1,  we  have 

F(u  +  1)  =  A(u  +  1)  u  (u  —  1)  .  .  .   (u  —  n  +  1)    multiplied  by 

(u  +  2)     (u  +  3)  .  .  .   (u  +  n  4-  1). 
It  follows  that 

II  I  -V)  1 

Phi  4.  n  —  EY?/)  u  "*"  n  ~  *-  • 

J.     \.li>       \^     A  J  A     \wy  • 

u  —  n 
or,  when  n  =  <*> , 

F(M  +1)=-  F(u). 

From  this  we  may  derive  at  once  the  relation 

sin  (u  +  n}  =  —  sin  w,     or     sin  (u  +  2  TT)  =  sin  w. 

ART.  21.     We  may  write 

sin  u  =  u  JJ  <  ( 1  —  — 

m      (  V  ^ 

where  the  product  extends  over  all  integers  m  =  ±  1,  ±  2,  ±  3,  .  .  .  , 
the  accent  over  the  product-sign  denoting  that  m  does  not  take  the  value 
zero.  u 

Owing  to  the  factor  e™*,  the  above  product  is  convergent  whatever  be 
the  order  of  the  factors. 

For  any  one  of  the  factors  [1 —  )eW3rmay  be  written 

V         mn) 


*  If  we  expand  the  sine-function  on  the  left  by  Maclaurin's  Theorem,  and  equate 
like  powers  of  u  on  either  side  of  the  equation,  it  follows  that  eG^  =  1. 


PRELIMINARY   NOTIONS.  19 

and  passing  to  the  product  of  such  terms  we  note  that  the  series 


are  absolutely  convergent. 

Since   m  takes  all  integral  values  from  -oo  to  +00  excepting   zero, 
we  may  change  the  sign  in  the  above  product  and  have 


sin  u  = 

Next  changing  u  to  —  u  and  comparing  the  resulting  product  with  the 
one  previously  derived,  we  see  that 

sin  (—  u)  =  —  sinw. 

The  point  u  =  GO  is  an  essential  singularity  of  sin  u.  For  if  we  put  u  =  - 
we  see  that  within  an  area  as  small  as  we  wish  about  v  =  0,  the  function 
sin  -  admits  an  infinity  of  zeros  v  =  —  ,  m  being  any  indefinitely  large 

integer.  It  follows  from  what  we  saw  in  Art.  3  that  i>  =  0orw=oois 
an  essential  singularity. 

ART.  22.     The  function  cot  u.  —  This  function  may  be  derived  from 
the  sine-function  from  the  formula 

cot  u  =  —  log  sin  u. 
du 

It  follows  from  the  above  formula  *  for  sin  u  that 


u 


1  1U/-J-         J_U 

.+  *        */      \U  +  2*        27T/ 


From  this  expression  we  have  at  once 

cot  (  —  u)  =  —  cot  u. 

We  also  note  that  the  points  0,  ±  -,   ±  2  n,  ...  are  simple  poles  and 
that  the  residue  with  respect  to  each  of  these  poles  is  unity. 
With  respect  to  any  of  these  poles,  say  u  =  TT,  the  difference 

cot  u  -- 

u  —  ~ 

is  a  regular  function  in  the  neighborhood  of  u  =  x. 

*  Eisenstein  (CreUe's  Journ.,  Bd.  35,  p.  191)  makes  use  of  this  formula  for  sin  u 
together  with  the  expression  for  cot  u  and  establishes  a  complete  theory  for  the  trigo 
nometric  functions. 


20  THEORY   OF   ELLIPTIC    FUNCTIONS. 

The  point  u  =  oo  is  an  essential  singularity. 
In  a  more  condensed  form  we  may  write 


where  the  summation  extends  over  all  integers  from  —  oo  to  +  oo  excepting 
zero. 

The  function 

sin2  u  du  u2        m    (u  —  mx)2 

is  an  even  function  which  has  as  double  poles  the  points  0,  ±  n,  ±  2  x,  -  •  •  . 
The  principal  part  relative  to  the  pole  u  =  mn  is 


(u  —  mil)2 

From  the  preceding  formulas  the  periodicity  of  the  circular  functions 
is  easily  established. 

The  expression  of  -  is  seen  to  remain  unchanged  when  n  is  added 
sin2  u 

to  u. 

For  the   cotangent   consider  the   difference   cot  (u  +  TT)  —  cot  u.    .  We 
find  that  the  expression 


/    i     _  I\  +  /!  _     i    \  +  /_!  ___  ! 

\U  +  71         Uj          \U         U  —  71  /          \U  —  7C         U  — 


—  2  7T/ 


/  _  L_          i    ^  +  /  _  i_        _1  _  \ 

\U  +  2  7T          U  +  71  1          U  +  37T          U  +  2n) 


is  zero. 

Further,  from  the  relation 

COt   (u   +  7t)   =  COt  U 

we  may  derive  the  periodicity  of  the  sine-function.     For  multiplying 
both  sides  of  this  expression  by  du  and  integrating,  we  have 
log  sin  (u  +  TT)  =  log  sin  u  +  log  C,     or 
sin  (u  +  TT)  =  C  sin  w. 

In  this  formula  put  it  =  —  -  ,  and  we  have  C  =  —  1. 

2i 

ART.  23.     Development  in  series.  —  If  we  note  that 


u  —  mn  mn       m27i2       mV 

it  is  seen  from  the  expansion  of  the  cotangent  that 

C0t«-!  -,!-?!-  ,„«?-„«£-    ... 

U  n2  7T4  7T6 


PRELIMINARY   NOTIONS.  21 


where  «i  =  2'-^,    s2  =  Y'l-     s3  =  X'^,   etc.      The  sums  V'-I, 
**  m2  **  Hi4  ^  w6  *~*  rra 

V' — ,  etc.,  are  evidently  zero,  since  the  positive  terms  are  destroyed 

^4    m3 

by  the  corresponding  negative  terms. 

To  determine  the  values  si,  s2,  .  .  .  ,  multiply  the  above  formula  by 
du  and  integrate. 

We  thus  have 

i  •  i  ,1  Si     U2          So  W4          SQ    U6 

log  sin  u  =  log  A  +  log  u  -  -±  —  -  -j  —  -  -^  ^  -  •  •  • 

_ Si  Jfr  _ S2  "4  _ 
or  sin  z*  =  AM  e    ^  *'     4  ff4 

Since    sm  u  =  1,    when   w   approaches    zero,    it   follows    that   A  =  1. 
Further,  since 


we  have  by  equating  like  powers  of  u,  after  the  exponential  function  on 
the  right  has  been  developed  in  series, 

=  *!  ~4  2;r6 

"  3  '    S2      32  -  5  '  33  •  5  •  7 ' 

(see  Bertrand's  Calcul  Differentiel,  p.  421). 
Noting  that 


n=l 

we  have  * 


22        32  66         2! 


24        34  90  30       4! 

11  -6  1          05  -6 

i  4  JL    _i_  _L    _(-...=      —  _  =  _  .       "•  , 

26        36  945  42        6! 

1        1     4-1     4..  -8  .J_.?L>rf 

28        38  9450  30       8! 

1    +    _L          _1_   4   .  7T10  5     §   297T10 

210      310  93555  66  '     10! 


The  numbers-,  — ,  — ,  — ,  — r  •  •  •   are  the  so-called  Bernoulli  num- 
6    30    42    30    66 

6ers  (cf.  Staudt,  Crelle's  Journ.,  Bd.  21,  p.  372). 

*  See    Biermann,    Theorie    der    analytischen    Functionen,   p.   326;    Jordan,    Traitt 
d' Analyse,  t.  I,  p.  360. 


22  THEORY   OF   ELLIPTIC    FUNCTIONS. 


THE  GENERAL  TRIGONOMETRIC  FUNCTIONS. 
ART.  24.     We  know  that  sin  2  u  =     2  cot  u         and 


cos  2  u  = 


1  +  cot2  u 
cot2  u  —  I 
cot2  w  +  1 


Further,  since  any  rational  function  of  a  trigonometric  function  may  be 
expressed  rationally  in  terms  of  the  sine  and  cosine,  we  may  consider  as 
the  general  case  any  rational  function  of  sin  u  and  cos  u  which  in  turn  is 

a  rational  function  of  cot-.  These  functions  remain  unchanged  when 
we  add  to  the  argument  u  any  positive  or  negative  multiple  of  2  n.  We 

say  that  2  n  is  a  primitive  period  of  these  functions.     Writing  cot  -  =  r, 

2i 

we  have  here  to  consider  any  rational  function  of  r.  Such  a  function  is 
consequently  a  one-valued  function  of  r  and  has  only  polar  singularities. 

As  in  the  case  of  the  rational  functions  we  shall  find  two  forms  for  the 
representation  of  the  trigonometric  functions,  the  one  corresponding  to 
the  decomposition  of  rational  functions  into  partial  fractions  and  the 
other  corresponding  to  the  expression  of  a  rational  function  as  a  quotient 
of  linear  factors. 

ART.  25.     First  form.  —  Write 

j,,^  =  F(sin  u,  cos  u) 
(r(sin  u,  cos  u) 

where  the  numerator  and  denominator  are  integral  functions  of  sin  u 
and  cos  u. 

Further,  since 

eiu  _  e-iu  eiu  _j_  g-tti 

sin  u  =  -     and     cos  u  =  -  —  -  , 
2i  2 

it  is  seen  that 
f(u\ 


B0(e2iu)n  +  Bite2''")"-1  +••••  +  Bn-ie2™  +  Bn  ' 

where  v  is  zero  or  is  an  integer  and  where  the  A's  and  B's  are  constants  or 
zero.  Through  division  we  may  express  f(u)  in  the  form  * 

f(u)  --=  P(eiu)  +  Q(eiu), 

where  P(eiu)  is  composed  of  integral  (positive  or  negative)  powers  of  eiu. 
But  in  Q(elu)  the  degree  of  the  numerator  is  not  greater  than  that  of  the 
denominator  and  this  denominator  does  not  contain  eiu  as  a  factor.  Hence 
Q(eiu)  =  $(u),  say,  remains  finite  when  u  =  <*>  and  also  when  u  =  —  oo  . 

*  Cf.  Hermite,  "Cours,"  loc.  cit.,  p.  121;  and  also  Hermite,  Cours  d'  Analyse  de  I'Ecole 
Poly  technique,  p.  321. 


PRELIMINARY   NOTIONS. 


23 


We  shall  next  study  the  function 
Consider  the  integral    /  $(u)du,  where  the  integration  is  taken  over 
the  contour  of  the  rectangle  ABCD  in  which 

OM  =  x0,  MN  =  2  TT,  AN  =  NB  =  a. 


Fig.  1. 


If  we  denote  by  (A B)  the  value  of  this  in 
tegral  taken  over  the  line  AB,  we  have  by 
Cauchy's  Theorem  (see  Art.  96) 

(AB)  +  (BC)  +  (CD)  +  (DA)  -  2-z'S, 

where  21  denotes  the  sum  of  the  residues  of 
<b(u)  corresponding  to  the  poles  that  are 
situated  on  the  interior  of  this  rectangle. 
Since  .r0  is  an  arbitrary  length,  the  sides  of 
the  rectangle  may  always  be  so  taken  that 


they  are  free  from  the  infinities  of  $(tf)- 

For  any  point  along  the  line  DC  we  may  write 

u  =  XQ+  IT, 

where  r  is  a  real  quantity  that  varies  from  —a  to  +a.     We  may  there 
fore  write 


(DC)    m  i     f+°  <*>(*<) 
J  -a 

r+a 
(AB)  =  i   I       &(x0 

J-a 

These  two  integrals  are  equal  since 


+  *T)<JT  and  similarly 


)  =  <&(u  +  2x).     It  follows  that 
(AB)  +  (CD)  =  0,     and  consequently 
(DA)  +  (BC)  =  2?-S;     or 


r 

Jo 


(XQ—  ia  +r)d 


C 
7   -    / 

Jo 


+  ia  +  r)dr  =  2  iVrS. 


(1) 


Next  let  the  constant  a  become  very  large  and  let  the  corresponding 
values  of 

®(XQ  -  ia  +r)  =  Q[e*"(*o-^+t)]  =  Q[e+a+iUo+T)      and 
$(xq  +  ia  +r)  =  Q  [e>(^+ta+T)]  =  Q[e 

be  respectively  G  and  H. 
Formula  (1)  becomes  then 

G  -  H  =  iS     or     2  =  T^ 


an  expression  which  gives  the  sum  of  the  residues  of  4>(u)  for  all  the  poles 
that  are  situated  between  the  parallels  AB  and  CD  when  indefinitely 
produced. 


24  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  apply  this  result  to  the  product 


Note  that 

-•"  -s   •    -         ei 


and  that  this  quantity  is  equal  to  —  i  for  u  =  <x>  and  to  +  i  for  u  =  —  oo . 
Hence  the  sum  of   the  residues  of   cotf    ~  u\$(u}  that  are   situated 

between  the  two  parallel  lines  above,  is  equal  to  —  G  —  H. 

We  may  next  compute  these  residues  and  equate  the  sum  of  the  residues 
computed  to  —  G  —  H. 

Let  the  poles  of  3>(u)  be  ai  of  order  ni, 

a2  of  order  n2, 


av  of  order  nv. 


We  know  that  the  residue  with  respect  to  a  pole  «i    is,  if  we  put 
=  u  —  a  i,  the  coefficient  of  -  in  the  dev 

it 

ing  increasing  powers  of  h  of  the  expression 


h  =  u  —  a  i,  the  coefficient  of  -  in  the  development  according  to  ascend- 

it 


By  Taylor's  Theorem 


=  cot  _          _  cot  _= 


2  2  II  dt  2 


(m  -  1)!  dPi~l  2 

Further,  the  expansion  of  ^>(ai  +  h)  in  the  neighborhood  of  ai  is  of  the 
form 


+  positive  powers  of  h. 
h        hz  hni 

If  we  put  A  =  Cn;   Ax  =  ^,  ^2=^;   •  •  •    ;    ^-1=  ^rf^  ^  is 

seen  that  the  coefficient  of  -  in  the  above  quotient  is 

h 

n         ±t  —  a\       „      d      ,  t  —  Oi    ,  ,   n      dni~l       .  t  —  a\ 


PRELIMINARY   NOTIONS.  25 

The  sum  of  the  residues  which  correspond  to  the  poles  of  <J>(u)  is  therefore 
represented  by 

d 


i      n     d     ^  t-di  ,  ,r     dn~       ^  t  -  a 

C2i-coi   _-  ,  ±0*00    -- 


2  "  dt  2 

Further,  with  respect  to  the  pole  u  =  t,  if  we  write  u  =  t  +  h  in  the 
quotient 

cos^-^ 


sn 


it  is  seen  that  the  coefficient  of  -,  when  h  is  very  small,  is  —  2 

/i 

We  thus  have 

-G-H  =  1CU  cot  -=-     -  C2i 


i=i  z 


or 


a  formula  which  is  similar  to  the  decomposition  of  a  rational  function  into 
its  simple  fractions  (see  Art.  11). 

ART.  26.  Second  form.  —  If  the  function  f(u)  becomes  zero  on  the 
points  ci,  c2  .  .  .  ,  cm  and  infinite  on  the  points  61?  62,  .  .  .  ,  bn,  it 
follows  at  once  from  the  expression  of  f(u)  above  that 


(e2iu-  e2ibl)  (e2iu—  e2ib2)  .  .  .  (e2iu—  e2l'6») 

=  Ae^ut'  sin  (M  ~  ci)  sin  (M  ~  ^2)  .  •  .  sni  (u  -  cw)  ^ 
sin  (u  —  61)  sin  (u  —  62)  .  .  .  sin  (M  —  6n) 

where  jj.  is  an  integer  and  C  and  A  are  constants. 

We  shall  see  later  (Arts.  373,  380)  that  there  are  analogous  representa 
tions  of  the  general  elliptic  function. 

REMARK.  —  The  functions  which  we  have  just  considered  admit  the 
period  2/r,  so  that 

f(u  +  2-)  =/(M). 


26 


THEORY    OF    ELLIPTIC    FUNCTIONS. 


If  we  change  the  variable  by  writing  u  =  — ,  so  that/(V)  =/(—-]  = 

CO  \  CO  / 

fi(u),  it  is  seen  that 

fi(u  +  2co)  =  fi  (u) , 

and  consequently  2  co  is  the  period  of  the  new  function;  and  further  all 

.  u 

Ttl  — 

rational  functions  of  eiu  are  now  rational  functions  of  e    w. 

In  the  next  Chapter  we  shall  show  that  any  trigonometric  function /(M) 
has  an  algebraic  addition-theorem;  or,  in  other  words,  f(u  +  v)  may  be 
expressed  algebraically  through  f(u)  and/(v). 


ANALYTIC   FUNCTIONS. 

ART.  27.  We  have  already  referred  to  certain  expressions  as  being 
analytic.  The  general  notion  of  an  analytic  function  may  be  'had  as 

follows.*  Suppose  that  the 
function  f(u)  has  a  finite  num 
ber  of  singular  points  pi,  p2, 
.  .  .  ,  pn  in  the  finite  portion  of 
the  w-plane.f 

From  each  of  these  points  we 
suppose  a  line  drawn  toward 
infinity,  the  only  restriction 
being  that  no  two  of  the  lines 
intersect  or  approach  each  other 
asymptotically.  {  These  lines 
we  may  consider  replaced  by 
canals  which  can  never  be 
crossed.  The  canals  we  suppose 
infinitesimally  broad,  so  that  all  the  points  of  the  w-plane  excepting  pi, 
p2,  .  .  .  ,  pn  are  either  on  or  outside  of  the  banks  of  the  canals,  the 
points  p  being  the  sources  from  which  the  canals  flow. 

We  suppose  that  the  function  f(u)  may  be  expanded  in  convergent 
power  series  in  positive  integral  powers  of  the  variable  at  all  points  except 

*  Weierstrass,  Abhand.  aus  der  Functionenlehre,  pp.  1  et  seq.;  Werke,  2,  p.  135.  See 
also  Vivanti-Gutzmer,  loc.  cit.,  pp.  334  et  seq.;  Goursat,  Cours  d' analyse,  t.  2;  Forsyth, 
Theory  of  Functions,  pp.  54  et  seq.;  Harkness  and  Morley,  Theory  of  Functions,  p.  105. 
Osgood  (Funktionentheorie,  p.  189)  defines  a  function  as  analytic  in  a  fixed  realm  when 
it  has  a  continuous  derivative  at  any  point  within  this  realm.  It  is  then  regular  at  all 
points  within  this  realm. 

t  We  have  supposed  the  function  defined  for  the  whole  plane;  it  may,  however, 
be  restricted  to  any  portion  of  this  plane. 

J  Mr.  Mittag-Leffler's  "star-theory"  suggests  that  the  plane  be  cut  so  as  to  have 
a  starlike  appearance  before  the  initial  Mittag-Leffler  star  is  formed.  See  references 
and  remarks  at  the  end  of  this  Chapter. 


PRELIMINARY   NOTIONS.  27 

Pi,  P2,  •  •  •  ,  Pn-  Let  a  be  any  such  point  and  let  P(u  —  a)  denote  the 
power  series  by  which  the  function  f(u}  may  be  represented  in  the  neigh 
borhood  of  a.  The  domain  of  the  absolute  convergence  of  this  series  is  a 
circle  having  a  as  center  and  with  a  radius  that  extends  to  the  nearest  of  the 
points  p  (see  Osgood,  loc.  cit.,  p.  285).  There  may  be  a  point  c  in  the 
w-plane  which  lies  without  this  domain  and  at  which  the  function  has  a 
definite  value.  The  function /(w)  may  also  at  c  be  expressed  in  the  form 
of  a  power  series  which  has  its  own  domain  of  convergence. 

The  question  is:  What  connection  is  there  between  the  two  power  series? 

Suppose  next  that  the  points  a  arid  c  are  connected  by  any  line  which 
does  not  cross  a  canal.  Take  any  point  a^  on  this  line  which  lies  within 
the  circle  of  convergence  about  a.  The  value  of  the  function  f(u)  at  'the 
point  ai  is  therefore  given  by  P(&i  —  a),  and  also  the  derivatives  of  f(u) 
at  the  point  ai  are  had  from  the  derivatives  of  this  power  series  after  we 
have  written  a±  for  u.  It  is  thus  seen  that  the  values  of  f(u)  and  of  its 
derivatives  at  a\  involve  both  a  and  a\. 

Next  draw  the  circle  of  convergence  about  ai  where  the  arbitrary  point 
«i  has  been  so  chosen  that  the  circle  about  a  and  the  circle  about  a\  inter 
sect  in  such  a  way  that  there  are  points  common  to  both  circles  and  also 
points  that  belong  to  either  circle  but  not  to  both. 

For  all  points  u  in  the  domain  of  a!  the  function/(&)  may  be  represented 
by  a  power  series,  say  P\(u  —  a\). 

We  may  show  as  follows  that  the  coefficients  of  this  power  series  involve 
both  a  and  a\\ 

For  the  domain  about  a  we  have  the  series 

(i)    /(«)  -/(a)  +  '^/»+  ("~a)V(«)  +  •  •  •  =  P(»  -«); 

and  for  points  common  to  the  domains  of  both  a  and  ai  we  have 

PI(U  -  a^  =  P(u  -  a)  =  P(a1  -  a  +  u  -  aj 

-  a)  +  P>!  _  a)  +  •  •  • 


In  the  domain  about  ai  we  have 

(II) 

where  in  this  domain 

/<*>(0l)  =  P<*>(a!  -  a),     /£  =  1,  2  .  .  .    , 

which  quantities  are  known  from  (I). 

Since  the  coefficients  of  PI(U  —  «i)  involve  both  a  and  a\,  the  power 
series  P\(u  —  ai)  is  sometimes  written  PI(U  —  ai,  a). 


28  THEORY    OF   ELLIPTIC    FUNCTIONS. 

At  a  point  u  situated  within  the  domains  common  to  both  a  and  a\  the 
two  series  P(u  —  a)  and  P\(u  —  ai)  give  the  same  value  for  the  function 
f(u).  Hence  the  second  series  gives  nothing  new  for  such  points.  But  for 
a  point  u  situated  within  the  domain  of  a±  but  without  the  domain  of  a, 
the  series  P\(u  —  a\,  a)  gives  a  value  of  f(u)  which  cannot  be  had  from 
P(u  —  a).  The  new  series  gives  an  additional  representation  of  the 
function.  It  is  called  a  continuation  *  of  the  series  which  represents  the 
function  in  the  initial  domain  of  a. 

Next  take  a  point  a2  situated  within  the  domain  of  «i  and  upon  the  line 
joining  a  and  c.  This  point  a2  is  to  be  so  chosen  that  its  domain  coin 
cides  in  part  with  the  domain  of  a\,  the  other  portion  of  the  domain  of  a2 
lying  without  that  of  a\.  The  values  of  f(u)  and  its  derivatives  at  a2  are 
offered  by  the  power  series  P\(u  —  &i,  a)  and  its  derivatives  when  for  u 
we  have  written  a2.  It  is  seen  that  for  all  points  common  to  the  domains 
a  i  and  0,2 

P2(u  —  o2)  =  PI(U  —  Oi)=  Pi(a2  —  0,1  +  u  —  a2) 


In  the  domain  about  a2 

(III)  /(«)  =  /(a2)  +1i^-«2/'(a2)+  (u~|a8)V(a)+.  .  .=  P2(«-a2), 
where  in  this  domain 


which  quantities  are  known  from  (II). 

It  is  thus  seen  that  the  coefficients  of  the  new  power  series  P2(u  —  a2) 
which  represents  f(u)  in  the  neighborhood  of  a2  involve  the  quantities  a  and 
a  i,  and  it  may  consequently  be  written  P2(u  —  a2)  =  P2(u  —  a2,  a,  d). 

At  those  points  u  in  the  domain  of  a2  which  do  not  lie  within  either  of 
the  two  earlier  circles  the  series  P2(u  —  a2j  a,  d)  gives  values  oif(u)  which 
cannot  be  derived  from  either  P(u  —  a)  or  P\(u  —  d)-  Thus  the  new 
series  is  a  continuation  of  the  older  ones. 

Proceeding  in  this  way  we  may  reach  all  the  points  of  the  w-plane  where 
the  function  behaves  regularly.  In  an  indefinitely  small  neighborhood  of 
those  points  p  which  are  essential  singularities  of  the  function  f(u),  the 

*  Weierstrass,  Werke,  Bd.  I,  p.  84,  1842,  employed  the  word  Fortsetzung;  MeYay, 
who  also  did  much  towards  the  foundation  of  the  theory  of  functions  by  means  of 
integral  power  series,  used  the  expression  cheminement,  a  series  of  circles  (see  M6ray, 
Leqons  nouvettes  sur  V  analyse  infinitesimale  et  ses  applications  gcomctriques.  Paris, 
1894-98). 


PKELIMINARY   NOTIONS.  29 

function  can  take  any  arbitrary  value  (Art.  3) ;  consequently  the  function 
may  be  continued  up  to  this  neighborhood  but  not  to  the  points  them 
selves;  while  it  may  be  continued  up  to  those  p's  which  are  polar  singu 
larities  (cf.  Stolz,  Allgenieine  Arithmetik,  Bd.  II.,  p.  100). 

The  combined  aggregate  of  all  the  domains  is  called  the  region  of  con 
tinuity  of  the  function.  With  each  domain  of  the  region  of  continuity  con 
structed  so  as  to  include  some  portion  not  included  in  an  earlier  domain,  a, 
series  is  associated  which  is  a  continuation  of  the  earlier  series  and  gives  at 
certain  points  values  of  the  function  that  are  not  deducible  from  the 
earlier  series.  Such  a  continuation  is  called  an  element  *  of  the  function. 
It  is  seen  from  above  that  any  later  element  may  be  derived  from  the 
earlier  elements  by  a  definite  process  of  calculation.  The  aggregate  of  all 
the  distinct  elements  is  called  an  analytic  function,  or  more  correctly  a 
monogenic  analytic  function,  the  word  monogenic  meaning  that  the  function 
has  a  definite  derivative.  As  only  functions  occur  in  the  present  treatise 
that  have  definite  derivatives,  the  word  monogenic  will  be  omitted  as 
superfluous. 

ART.  28.  We  may  note  that  there  are  functions  which  although  finite 
and  continuous  have  no  definite  derivatives.  Weierstrass  (Crelle's  Journ., 
Bd.  79,  p.  29;  Werke,  Bd.  II.,  p.  71)  shows  this  by  means  of  the  function  t 

f(u)  =  2an  cos  bnu, 

which,  although  always  finite  and  continuous,  never  has  a  definite  deriva 
tive,  if  b  is  an  odd  integer  and 

(1st)    ab  >  1  +  |  T.     or     (2d)    ab2  >  1  +  3  -2, 
where  in  the  first  case  ab  >  1  and  in  the  second  case  ab  must  be  =  1. 

ART.  29.  If  c  is  any  point  hi  the  region  of  continuity  but  not  neces 
sarily  in  the  circle  of  convergence  of  the  initial  element  about  a,  it  is  evi 
dent  that  a  value  of  the  function  at  c  may  be  obtained  through  the  con 
tinuations  of  the  initial  element.  In  the  formation  of  each  new  domain 
(and  therefore  of  each  new  element)  a  certain  amount  of  arbitrary  choice  is 
possible;  and  as  a  rule  there  may  be  different  sets  of  domains  (for  example 
in  the  figure  of  p.  26  along  another  path  abi  62  .  .  .  c),  which  domains 
taken  together  in  a  set  lead  to  c  from  the  initial  point  a.  So  long  as  we  do 
not  cross  a  canal  and  consequently  do  not  encircle  any  of  the  singular 
points  p,  the  same  value  of  the  function  at  c  is  had,  whatever  be  the  method 
of  continuation  from  the  initial  point  a.  The  function  is  one- valued  in  the 
plane  where  the  canals  have  been  drawn. 

*  Weierstrass,  Werke  2,  p.  208. 

t  See  also  Jordan,  Traite  d' Analyse,  t.  3,  p.  577;  Dini,  Fondamenti  per  la  teorica 
delle  funzioni  di  variabili  reali,  §  126;  Wiener,  Crelle,  Bd.  90,  p.  221;  Picard,  Traite 
d' Analyse,  t.  2,  p.  70;  Forsyth,  Theory  of  Functions,  p.  138;  Hadamard's  Thesis,  Journ. 
de  Math.,  1872;  Darboux,  Memoire  sur  I' approximation,  etc.,  Liouv.  Journ.,  1877; 
Osgood,  Lehrbuch  der  Funktionentheorie,  p.  89;  Pringsheim,  Ency.  der  Math.  Wiss., 
Bd.  II,1  Heft  1,  pp.  36  et  seq. 


SO  THEORY   OF   ELLIPTIC    FUNCTIONS. 

In  Chapter  VI  it  will  be  seen  that  if  the  crossing  of  a  canal  is  allowed  we 
may  have  different  values  of  the  function  at  c;  in  fact,  the  function  has  at 
c  just  as  many  values  *  as  there  are  different  elements  P(u  —  c)  which 
lead  back  to  the  same  initial  element  at  a. 

ART.  30.  The  whole  process  given  above  is  reversible  when  the  function 
is  one-valued.  We  can  pass  from  any  point  to  an  earlier  point  by  the  use 
if  necessary  of  intermediate  points.  We  thus  return  to  the  point  a  with  a 
certain  functional  element,  which  has  an  associated  domain.  From  this 
the  original  series  P(u  —  a)  can  be  deduced.  As  this  result  is  quite  general, 
any  one  of  the  continuations  of  a  one-valued  analytic  function  repre 
sented  by  a  power  series  can  be  derived  from  any  other;  and  conse 
quently  the  expression  of  such  a  function  is  potentially  given  by  any  one 
element.  This  subject  is  treated  more  fully  in  Chapter  VI. 

To  effect  the  above  representation  of  an  analytic  function  it  is  often 
necessary  to  calculate  a  number  of  analytic  continuations,  for  each  of 
which  we  must  find  the  radius  of  the  circle  of  convergence.  Thus  (cf. 
also  Mr.  Mittag-Leffler,  f  one  of  the  greatest  exponents  of  Weierstrass's 
Theory  of  Functions)  it  is  seen  that  the  manner  given  above  of  repre 
senting  a  function  by  means  of  its  analytic  continuations  is  an  extremely 
complicated  one.  It  seems  that  Weierstrass  scarcely  regarded  the  ana 
lytic  continuation  other  than  as  a  mode  of  definition  of  the  analytic  func 
tion.  As  a  definition  it  has  great  advantages. 

But  the  theory  of  Cauchy  (cf.  again  Mittag-Leffler),  which  is  founded 
upon  quite  different  principles,  has  in  most  other  respects  greater  advan 
tages. 

The  representation  of  a  function  by  means  of  the  integral 


-  £.  f  JfiUt 

2mJsz  —  u 


the  integration  being  taken  over  a  closed  contour  S  situated  within  the 
region  for  which  f(u)  is  defined,  is  fundamental  in  the  derivation  of 
Taylor's  Theorem  for  a  function  of  the  complex  argument. 

Mr.  Mittag-Leffler  $  gives  an  extension  of  Taylor's  Theorem  in  his 
"  star-theory  "  by  means  of  which  he  treats  the  "  prolongation  of  a 
branch  of  an  analytic  function  "  in  a  very  comprehensive  manner. 

General  methods  of  representing  an  analytic  function  in  the  form  of 

*  Vivanti  (see  Vivanti-Gutzmer,  loc.  cit.,  p.  109)  gives  a  method  by  which  a  many- 
valued  function  may  be  considered  as  a  combination  of  one-valued  functions.  See  also 
Weierstrass,  Abel'sche  Transcendenten,  Werke,  4,  p.  44. 

In  the  sequel  we  shall  by  means  of  canals  so  arrange  our  plane  or  surface  on  which 
the  function  is  represented,  that  the  function  may  be  always  regarded  as  one-valued. 

f  Sur  la  representation  analytique,  etc.,  Acta  Math.,  Bd.  23,  p.  45. 

J  Mittag-Leffler,  Sur  la  representation,  etc.,  Seconde  note,  Acta  Math.,  Bd.  24,  p.  157; 
Troisieme  note,  Acta  Math.,  Bd.  24,  p.  205;  Quatrieme  note,  Acta  Math.,  Bd.  26,  p.  353; 
Cinquieme  note,  Acta  Math.,  Bd.  29,  p.  101. 


PRELIMINARY   NOTIONS.  31 

an  arithmetical  expression  are  given  by  Hilbert,  Runge,  and  Painleve 
(see  Vivanti-Gutzmer,  loc.  cit.,  pp.  349  et  seq. ;  Osgood,  Encyklopddie  der 
Math.  Wiss.,  Bd.  II2,  Heft  1,  pp.  80  et  seq.}. 

ART.  31.  Algebraic  addition-theorems. — We  have  seen  that  the 
rational  functions  are  characterized  by  the  properties  of  being  one-valued 
and  of  having  no  other  singularities  than  poles.  These  functions  possess 
algebraic  addition-theorems. 

We  have  also  seen  that  the  general  trigonometric  functions  (rational 
functions  of  sin  u  and  cos  u  or  of  cot  u/2)  have  only  polar  singularities  in 
the  finite  portion  of  the  plane.  These  functions  have  periods  which  are 
integral  multiples  of  one  primitive  period  2  TT.  These  properties,  however, 
do  not  characterize  the  trigonometric  functions;  for  they  belong  also  to 
the  function  esinu  which  is  not  a  trigonometric  function.  To  character 
ize  the  trigonometric  functions,  it  is  necessary  to  add  the  further  con 
dition  that  they  have  algebraic  addition-theorems,  as  is  shown  in  the  next 
Chapter. 

We  shall  call  an  elliptic  function  *  a  one-valued  analytic  function  which 
has  only  polar  singularities  in  the  finite  portion  of  the  plane  and  which 
has  periods  composed  of  integral  (positive  or  negative)  multiples  of  two 
primitive  periods,  say  2  a>  and  2  a/;  for  example, 

f(u  +  2  a>)  =  f(u),  f(u  +  2  a/)  =  f(u) 

and  f(u  +  2  mw  +  2  W)  =  f(u), 

where  m  and  n  are  integers. 

A  further  condition  is  that  these  functions  have  algebraic  addition- 
theorems.  Weierstrass  characterized  as  an  elliptic  function  any  one-val 
ued  analytic  function  as  defined  above  which  has  only  polar  singularities 
in  the  finite  portion  of  the  plane  and  which  possesses  an  algebraic  addi 
tion-theorem,  the  trigonometric  functions  being  limiting  cases  where  one 
of  the  primitive  periods  becomes  infinite,  as  are  also  the  rational  func 
tions  which  have  both  primitive  periods  infinite. 

EXAMPLES 
1.    Prove  that 


where  m  takes  all  integral  values,  negative,  zero,  and  positive. 
2.   Show  that 


Vr°  \(i !L_y£iJ  =  sin  x(u  +  a)  e. 

m= -a,  (  \         m  -  a)  )  sin  -a 

*  To  be  more  explicit,  such  a  function  is  an  elliptic  function  in  a  restricted  sense. 
The  more  general  elliptic  functions  include  also  the  many- valued  functions  (see  Chapter 

WTA 


32  THEOKY   OF   ELLIPTIC    FUNCTIONS. 


3.   Show  that 

m=  +00 


m=  —oo 

4.   Show  that 


6.  Show  that  [GaUSS'] 

1  3  COS  71X 


(x  +  w)3  sin 

( 


and  that 


1         _  _4  /     2       1  1     \ 

a;  +  m)4  \     3  sin2  TTZ      sin4  TTO;/' 


(2  0,  x)  =  V * =  ^gf     at      +      <*2 

^^Crr  4-  w.'l2^  «in2  TTV      oir.4  ^^ 


+  m)2<7  sin2  ^     sin4  ^  sin2* 


3  5 


(2g 

sin   ^        sin 

where  the  coefficients  a1;  a,,  .  .  .   ;  6lf  62,  .  .  .   are  connected  with  the  Bernoulli 
numbers  in  a  simple  manner  and  may  be  found  by  successive  differentiation. 

Eisenstein,  Crelle,  Bd.  35,  p.  198; 

Euler,  Introductio  in  analysin  infinitorum. 
6.   Prove  that 

3  (4,  x)  =  (2,  x)2  +  2(1,3)  (3,3); 
3(2,  0)  =  7T2. 


CHAPTER   II 
FUNCTIONS    WHICH    HAVE    ALGEBRAIC    ADDITION-THEOREMS 

Characteristic  properties  of  such  functions  in  general.     The  one-valued  functions. 
Rational  functions  of  the  unrestricted  argument  u.     Rational  functions  of  the 

iciu 

exponential  function  e  w  . 

ARTICLE  32.     The  simplest  case  of  a  function  which  has  an  algebraic 
addition-theorem  is  the  exponential  function 

<t>(u)=  eu. 
It  follows  at  once  that 

eu  +  v=  eu  -ev, 


or  <j>(u  +  v)=  (j) 

Such  an  equation  offers  a  means  of  determining  the  value  of  the  function 
for  the  sum  of  two  quantities  as  arguments,  when  the  values  of  the  func 
tion  for  the  two  arguments  taken  singly  are  known. 

It  is  called  an  addition-theorem. 

In  the  example  just  cited  the  relation  among  $(M),  <j>(v)  and  <j)(u  +  v) 
is  expressed  through  an  algebraic  equation,  and  consequently  the  addi 
tion-theorem  is  called  an  algebraic  addition-theorem. 

The  theorem  is  true  for  all  values  of  u  and  v,  real  or  complex.  The 
exponential  function  eu  is  perhaps  best  studied  by  deriving  its  properties 
from  its  addition-theorem. 

The  sine  function  has  the  algebraic  addition-theorem 

sin  (u  +  v)  =  sin  u  cos  v  +  cos  u  sin  v 

=  sin  u  \/l  —  sin2  v  +  sin  v  Vl  —  sin2  u. 

The  root  signs  may  be  done  away  with  by  squaring. 
We  also  have     '  tan  u  +  tan  ^  ^ 

1  —  tan  u  tan  v 

We  note  in  the  above  algebraic  addition-theorems  that  the  coefficients 
connecting  <j>(u),  <j>(v),  and  <j)(u  +  v)  are  constants,  that  is,  quantities 
independent  of  u  and  v. 

With  Weierstrass  *  the  problem  of  the  theory  of  elliptic  functions  is  to 

*  Cf.  Schwarz,  Formeln  und  Lehrsdtze  zum  Gebrauche  der  elliptischen  Functionen, 
pp.  1  et  seq.  The  Berlin  lectures  of  Prof.  Schwarz  have  been  of  service  in  the  prepa 
ration  of  this  Chapter. 

38 


34  THEORY   OF   ELLIPTIC    FUNCTIONS. 

determine  all  functions  of  the  complex  argument  for  which  there  exists  an 
algebraic  addition-theorem. 

Every  function  for  which  there  exists  an  algebraic  addition-theorem 
is  an  elliptic  function  or  a  limiting  case  of  one,  those  limiting  cases  being 
the  rational  functions,  the  trigonometric  and  the  exponential  functions. 

ART.  33.     We  may  represent  a  function  of  the  complex  argument  by 


and  further  we  shall  write 


We  may  assume  either  that  the  function  (f>  is  defined  for  all  real, 
imaginary,  and  complex  values  of  the  argument,  or  that  this  function  is 
denned  for  a  definite  region,  which,  however,  must  lie  in  the  neighbor 
hood  of  the  origin.  Further  it  is  assumed  that  (f>  has  an  algebraic  addition- 
theorem.  We  therefore  have,  if  G  represents  an  integral  function  with 
constant  coefficients, 

G(t,  1,  0  =  0. 

We  may  now  derive  other  properties  of  such  a  function  from  the  property 
that  there  exists  an  algebraic  addition-theorem. 

ART.  34.  If  we  differentiate  the  function  G  with  respect  to  u,  then, 
since  £  is  independent  of  77,  we  have 

^£+^£  =  0,  and  similarly 
d£  du       d    du 


dif]  dv       d£  dv 

Write  u  +  v  =  h  and  note  that  fC  »££•  1  _«'££ 

du       dh  dv 

We  consequently  have  by  subtraction 

d 
dv 


There  are  two  cases  possible  : 

dC1    dC1 

First.     The  quantity  r  may  appear  in  the  coefficients  —  >  —  ;  or 

3£    drj 

Second.     The  quantity  £  does  not  appear  in  these  coefficients. 
Consider  the  first  case.     We  have  the  two  equations 

G(?>  1,  0  =  0, 


du       difj  dv 


ALGEBRAIC   ADDITION-THEOREMS.  35 

The  first  of  these  equations  may  be  written 


where  the  a's  are  integral  functions  of  £  and  y;  the  second  equation  may  be 
written 


where  the  A's  are  integral   functions  of  £,  fl,  —  and  ^  .     If  ^  is  elimi- 

du  dv 

nated  from  these  two  expressions,  we  have 


In  the  second  case  where  £  does  not  appear  in  the  coefficients—  and 

d? 

-  ,  we  have  at  once  an  equation  connecting  ?.—-,•  and  ^  . 
°ri  du  dv 

This  case  is,  however,  the  very  exceptional  one.  We  have  by  the 
above  considerations  put  into  evidence  a  new  property  of  the  function  <£, 
viz.: 

//  the  function  (f>  has  an  algebraic  addition-theorem,  there  is  always  an 
equation  of  the  form 


where  H  represents  an  integral  function  of  its  arguments  with  constant  coeffi 
cients.  The  equation  is  true  for  all  values  of  u  and  v  which  lie  within  the 
ascribed  region. 

This  equation  being  true  for  all  such  values  of  v,  we  may  give  to  v  a 

j,ju 

special  value,  and   have  consequently  between  c  and  —  an  equation  of 

du 
the  form 


where  /  denotes  an  integral  function  of  its  arguments. 

This  equation  we  shall  call  the  eliminant  equation*     We  may  write  it 
in  the  form 

,     »    -  0. 


We  have  therefore  proved  that  if  for  the  analytic  function  (j>(u)  there 
exists  an  algebraic  addition-theorem,  we  also  have  an  algebraic  equation 
between  the  function  and  its  first  derivative,  the  equation  being  an  ordinary 

*  The  equation  is  due  to  Meray,  see  Briot  et  Bouquet,  Theorie  des  Fonctions  Ellip- 
tiques,  p.  280;  Picard,  Traitc  d'Analyse,  t.  2,  p.  510;  Daniels,  Amer.  Journ.  Math., 
Vol.  VI,  pp.  254-255. 


36  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

differential  equation  of  the  first  order.  The  argument  u  does  not  appear 
explicitly  in  the  equation. 

ART.  35.  As  the  above  theorem  is  made  fundamental  in  many  of  the 
following  investigations,  it  is  of  great  importance  to  note  that  it  is  true 
without  exception. 

-  In  the  equation  H  =  0  we  may  write  any  arbitrary  value  for  v  which 
belongs  to  the  region  considered.     If  after  the  substitution  of  this  value 

of  v  there  remains  an  equation  between  £  and  — ,  then  our  conclusions 

du 

above  are  correctly  drawn;  but  if  after  the  substitution  of  this  value  of 
v  the  equation  were  to  vanish  in  all  its  coefficients,  the  theorem  remains 
yet  to  be  established.  We  take  the  following  method  to  prove  that  the 
theorem  is  always  true: 

Develop  the  function  H  in  powers  of  £  and  —  .     The  coefficients  in  this 

du 

development  are  either  zero  or  functions  of  in  and  -2 1  including  constants. 

dv 

It  is  evident  that  all  of  the  coefficients  are  not  zero,  for  then  the  function 
H  would  be  identically  zero. 

We  represent  one  of  the  coefficients  which  is  not  zero  by 


There  must  be  such  a  coefficient  which  contains  in  and  -2  ;  for  otherwise 

dv 

all  the  coefficients  would  be  independent  of  these  quantities,  which  there- 

aXTj 

fore  would  not  enter  the  function  H.     But  since  —    is  not  always  zero, 

677 

these  quantities  must  appear. 

In  this  coefficient /i  (77,  — -*  j  we  give  v  a  definite  value,  and  if  the  value 

resulting  of  the  coefficient  is  different  from  zero,  then  in  the  above  devel 
opment  we  have  an  equation  connecting  £  and  —  . 

du 

But  if  this  value  of  v  causes  fji),  ~^j  to  be  zero,  we  try  another  value 

and  continue  until  we  find  a  value  of  v  that  causes  this  coefficient  to  be 
different  from  zero,  if  this  be  possible. 

If,  however,  the  function /if  n,  -5  ]  is  zero  for  every  value  of  v,  we  have 

V     dv/ 
an  equation  of  the  form 

/i0W,  </>»]  -  0, 

where  f\  is  an  integral  function  of  its  arguments.     This  equation,  how 
ever,  expresses  the  same  thing  as  the  equation 

,  </>'(")]  -  0, 


ALGEBRAIC  ADDITION-THEOREMS.  37 

only  in  the  first  case  the  argument  is  v  and  not  u,  which  of  course  makes 
no  difference. 

If  any  of  the  coefficients  in  the  development  of  the  function  H  con 
tained  7)  alone,  /2( TJ)  being  such  a  coefficient,  then  since /2  is  an  integral 
function  of  finite  degree  it  can  vanish  only  for  a  finite  number  of  values  of 
7),  and  we  have  only  to  give  TJ  a  value  such  that/2(^)  7^  0. 

The  theorem  is  therefore  true  without  exception  for  every  analytic 
function  for  which  there  exists  an  algebraic  addition-theorem  with  con 
stant  coefficients;  and  conversely,  as  will  be  shown  in  Chapters  VI  and  VII, 
if  a  one-valued  analytic  function  <j>(u)  has  the  property  that  between  the 
function  <f>(u)  and  its  first  derivative  <j>'(u)  there  exists  an  algebraic  equation 
whose  coefficients  are  independent  of  the  argument  u,  the  function  has  an 
algebraic  addition-th eorem. 

This  eliminant  equation  (see  also  Forsyth,  Theory  of  Functions,  p.  309) 
must  be  added  as  a  latent  test  to  ascertain  whether  or  not  an  algebraic 
equation  connecting  £,  y,  £  is  one  necessarily  implying  the  existence  of  an 
algebraic  addition-theorem.  We  must  not  suppose  that  every  algebraic 
equation 

G(^  i,  0=0 

necessarily  exacts  the  existence  of  an  algebraic  addition-theorem;  neither 
does  the  relation 

<t>(u  +  v)  =  F{<l>(u),  4>'(u),  <f>(v),  <t>'(u)}, 

where  F  denotes  a  rational  function  of  its  arguments,  always  indicate  the 
existence  of  such  a  theorem.     (See  Art.  46.) 
ART.  36.     If  we  solve  the  equation 


/— 

with  respect  to  —  ,  we  have 
du 


f-*®- 


where  ^(£)  is  an  algebraic  function  of  £. 
This  equation  may  be  written 


or  .....       <fe 


f1 

u  —  UQ  =  I 

Ja 


where  I^Q  and  a  denote  constants. 

It  is  thus  seen  that  in  the  case  of  every  analytic  function  £  =  <j>(u),  for 
which  there  exists  an  algebraic  addition-theorem  with  constant  coefficients^ 
the  quantity  u  may  be  expressed  through  the  integral  of  an  algebraic  func 
tion  of  £. 


38  THEORY  OF  ELLIPTIC  FUNCTIONS. 

We  may  so  choose  the  initial  value  a  that  u0  =  0,  thus  having 


In  a  similar  manner 

» 

On  the  other  hand  we  have 


We  thus  have  the  equation 


a  formula  which  is  of  fundamental  importance. 

To  illustrate  the  significance  of  the  above  formula  consider  the  follow 
ing  examples: 

1.    Let  e  =  <f>(u)  =  eu;  <j>f(u)  .  eu. 

We  therefore  have  as  the  eliminant  equation 

«-?• 

rfw 

and  also  ^(l")  =  <f 

Since  ^  =  1  when  a0  =  0,  we  may  write 

' 


On  the  other  hand,  £  =  0(M  +  v)  =  eu+v=  eu-ev=  </>(u)  6(v)  =  £  .  »      It 
follows  that 


t       ,  t 

or  log  £  +  log  ^  =  log  t.y. 

2.    Let  f  =  <f>(u)  =  sin  w;  0/(?/)  =  cos  w  =  Vl  -  sin2  w  -  Vl  -  ^2. 
It  follows  that  i/r(f)  =  x/1  -  <f2,  and  consequently  since  w  =  0  for  £  =  0 


'o  VI  -  t2 
Further,  since 

we  have 


'o  v/1  -  t2     Jo  Vl  -  t2      Jo       Vl  - 
or  sin-1*?  +  sin-1  y  =  sin-1  [£  Vl  -  -rf  +  v  V 


ALGEBRAIC  ADDITION-THEOREMS.  39 

3.    If  £  =  tan  u  =<j>(u),  we  have 

l±i 
p    dt          p     dt          p-fr    dt 

Jo  1  +  t2     Jo  1  +  t2     Jo        1  +  *2' 

or  tan  - l  £  +  tan  ~ l  y  =  tan  ~ 1    *  ^  J'     • 

l_l  ~~  ^7/J 

ART.  37.  We  have  seen  that  for  every  function  £  =  <£(M)  for  which 
there  exists  an  algebraic  addition-theorem,  there  exists  without  excep 
tion  a  differential  equation  of  the  form 


/[</>(«),  </,»]  =  0,  or/(f,£)=0, 


where  /  denotes  an  integral  function  of  its  arguments  and  where  u  does 
not  appear  explicitly  in  the  equation. 

If  c  =  <f>(u)  is  known  for  a  definite  value  of  u,  then  from  the  above 

si- 

equation  we  ma}7  determine  ^-,  there  being  one  or  more  values  according 

du 

to  the  degree  of  the  equation  in  -^   . 

du 

We  may  now  prove  the  following  theorem:  //  the  function  £  =  <j>(u)  has 
an  algebraic  addition-theorem,  the  values  of  all  the  higher  derivatives  of 
<f>(u)  with  respect  to  u  may  be  expressed  as  rational  functions  with  constant 
coefficients  of  the  function  itself  and  its  first  derivative;  so  that  if  the  values 
of  the  function  and  its  first  derivative  are  known,  the  higher  derivatives  are 
uniquely  determined. 

There  are  exceptions  to  the  theorem  which  are  noted  in  the  following 

proof:   If  we  write  —  =  £',  the  equation  above  becomes 
du 

/(£,  £')  =  0,     or,  say, 


where  n  is  a  positive  integer  and  the  a's  are  integral  functions  of  £. 

We  may  assume  that  /(£,  £')  is  an  irreducible  function,  that  is,  it  cannot 
be  resolved  into  two  integral  functions  of  £,  £';  for  if  this  were  the  case, 
one  of  the  factors  put  equal  to  zero  might  be  regarded  as  the  integral 
equation  connecting  £  and  £'. 

We  form  the  derivative    ^  ^^    ,  which  is  an  integral  function  in  £,  £'. 

The  degree  of  this  derivative  in  £'  is  one  less  than  the  degree  of  /(£,  £') 
in  £'. 

Further,  the  equation  ^  ^\,  =  0  is  not  satisfied  for  all  pairs  of  values 
£,  ^'  which  satisfy  the  equation  /(£,  cr)  =  0.  For  if  this  were  the  case, 


40  THEORY    OF    ELLIPTIC    FUNCTIONS. 

the  two  equations  would  have  a  greatest  common  divisor,  this  divisor 
appearing  as  a  factor  of  both  functions.  But  by  hypothesis  /(£,  £')  is 
irreducible.  The  two  equations 

/(£,  f)  =  o, 


are  satisfied  by  only  a  finite  number  of  pairs  of  common  values  £,  if'. 
For  their  discriminant  with  respect  to  £'  is  an  integral  function  in  the 
a's;  and  as  this  discriminant  put  equal  to  zero  is  the  condition  of  a  root 
common  to  both  equations,  we  have  an  integral  equation  in  the  a's,  that 
is,  in  £.  There  are  consequently  only  a  finite  number  of  values  of  £  which 
satisfy  this  condition. 

These  common  roots  constitute  the  exceptional  case  mentioned  at 
the  beginning  of  the  article  and  are  excluded  from  the  further  investi 
gation.  They  may  be  called  the  singular  roots. 

We  next  consider  a  value  u  =  UQ  of  the  argument,  for  which  </>(UQ)  =  £o> 
<t>'(uo)  =  £o',  where  £o>  £o'  satisfy  the  equation  /(£,  £')  =  0  but  not  the 


equation  =  0. 

d$ 

By  differentiation  we  have 


We  further  assume  that  the  point  in  question  is  such  that  the  function 
has  for  it  a  definite  derivative. 
We  may  write 


d?  =  ?'du. 
It  then  follows  that 


or 


u 

a?' 


From  this  it  is  seen  that  f"  =  0"(u)  is  rationally  expressed  through 
f,   f.     Since  the  singular  roots  have  been  excluded,  the  denominator 


In  a  similar  manner  it  may  be  shown  that  £'"  =  <j)fff(u')  may  be  expressed 
in  the  form  of  a  fraction  whose  denominator  is  a  power  of  the  denominator 


ALGEBRAIC  ADDITION-THEOREMS.  41 

which  appears  in  the  expression  for  £"  and  consequently  is  different  from 
zero.     The  same  is  true  for  all  higher  derivatives. 

ART.  37a.     Suppose  that  u\  is  a  value  of  the  argument  u  different  from 
UQ  and  such  that 

<£  (U0)    =  CO   =  <t>(Ui), 


Further  let  <}>(u)  be  an  analytic  function  with  an  algebraic  addition- 
theorem,  and  in  the  neighborhood  of  UQ  and  u\  let  the  function  (f>(u)  be 
regular.  Finally,  it  is  assumed  that 


that  is,  £V  does  not  belong  to  the  singular  roots  of  /(c,  £')  =  0. 

We  assert  that  <f>(u)  under  these  conditions  is  a  periodic  function  and  that 
UI  —  UQ  is  a  period  of  the  argument.* 

Since  the  function  <f>(u)  is  regular  in  the  neighborhood  of  UQ,  it  may  be 
developed  by  Taylor's  Theorem  in  the  form 


<t>(uQ)  +  - 
In  a  similar  manner  we  also  have 


By  hypothesis  we  have 


The  derivative  0"  (MO)  may  be  expressed  as  a  rational  function  of  (/)(UQ), 
<J>'(UQ)  with  constant  coefficients;  (/>"(ui)  has  the  same  form  in  </>(MI), 


It  follows  that 

$"  (UQ)  =  (f>"(u\),  and  in  a  similar  manner 


Let  MQ  +  v  be  a  point  that  lies  within  the  region  of  convergence  of  the 
first  of  the  above  series  and  let  HI  +  v  be  a  point  situated  within  the 
region  of  convergence  of  the  second. 

*  Cf .  Biennann,  Theorie  der  analytischen  Funktionen,  p.  392. 


42  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Instead  of  u  write  u0  +  v  and  u\  +  v  in  the  two  series  respectively.    They 
become 


Consequently,  owing  to  the  relations  above, 

<f>(UQ   +  V)=<l>(Ui   +  V). 

Next  write  Ui  —  u0  =  2  at     or     u\  =  u0  +  2w,  and  we  have 
0(^o  +  v)  =  (f>(u0  +  v  +  2a>). 

The  quantity  v  may  be  regarded  as  an  arbitrary  complex  quantity, 
and  must  satisfy  the  condition  that  UQ  4-  v  belongs  to  the  region  for  which 
</>(u)  has  been  defined. 

The  quantity  2  a>  is  called  the  period  of  the  argument  of  the  function,  less 
accurately  the  period  of  the  function. 

We  may  therefore  conclude  that  a  function  <f>(u)  is  periodic,  if  it  has  an 
algebraic  addition-theorem  and  if  there  are  two  points,  UQ  and  u\,  that 
are  not  the  singular  roots  of  f[(f>(u),  <£'(»]=  0,  for  which 

4>(uQ)  *~  <f>(tii)     and     <J>'(UQ)=  <t>'(u\)> 

ART.  38.  If  we  have  only  the  one  condition  that  <J>(UQ)  =  $(MI),  we 
cannot  without  further  data  draw  the  same  conclusions  about  periodicity. 
If  the  equation  connecting  </>(u)  and  <t>'(u)  is  of  the  first  degree  in  <f>'(u), 
as  is  the  case  of  the  exponential  function,  then  the  second  condition, 
viz.,  <p'(uo)  =  (t>'(ui)  follows  at  once.  In  general  this  is  not  the  case. 

We  may,  however,  effect  a  conclusion  if  the  assumptions  are  somewhat 
changed:  Suppose  that  n  is  the  degree  of  the  equation  f[$(u),  <f>'(u)]  =  0 
with  respect  to  <f>'(u).  To  every  value  of  <f>(u)  there  belong  at  most 
n  values  of  </>'(u). 

Suppose  next  that  n  +  1  points  UQ,U\,  .  .  .  ,  un  may  be  found,  at 
which 


and  suppose  also  that  <fr(u)  is  regular  in  the  neighborhood  of  each  of  these 
points,,  and  further  suppose  that  £0  is  not  a  singular  root  of  /(£,  £')  =  0. 
Write 

<J>'(U0)=   ttfo, 


<t>'(un)=  CJn. 

These  n  +  1  values  of  <f>'(u)  belong  to  one  value  of  £0  =  <t>(uo)  =  <i>(u\)  = 
.  .  .  =  (j)(un).     But  as  there  can  only  be  n  values  of  (j>'(u)  belonging  to  one 


ALGEBRAIC   ADDITION-THEOREMS.  43 

value  of  <j>(u),  it  follows  that  two  of  the  above  values  of  (j>'(u)  must  be 
equal,  and  consequently 

^(ti«)  -£'(*,), 

where  a  and  ft  are  to  be  found  among  the  integers  0,  1,  2,  .  .  .  ,  n.     But 
by  hypothesis  we  also  had 


It  follows  from  the  theorem  of  the  preceding  article  that  <f>(u)  is  periodic, 
u*  —  up  being  a  period  of  <t>(u).     We  have  then  the  following  theorem:  * 

//  it  can  be  shown  that  a  function  having  an  algebraic  addition-theorem 
takes  the  same  value  on  an  arbitrarily  large  number  of  positions  in  the  neigh 
borhood  of  which  the  function  is  regular,  the  function  is  periodic. 

ART.  39.     We  have  seen  that  in  the  equation  connecting  £  and  —  ,  viz., 

du 


the  quantity  u  does  not  explicitly  appear. 

Suppose  that  ~  =  <f>(u)  is  a  particular  solution  of  this  differential  equa 
tion.  As  this  differential  equation  is  of  the  first  order,  the  general  solution 
must  contain  one  arbitrary  constant. 

We  may  introduce  this  constant  by  writing 


the  arbitrary  constant  v  being  added  to  the  argument.     It  makes  no 
difference  whether  we  differentiate  with  regard   to  u  or  with  regard   to 
u  +  v  since  u  does  not  enter  the  equation  explicitly. 
We  consequently  have 


I// 

~  *  (u 


from  which  it  is  seen  that  the  differential  equation  is  satisfied  by  </>(u  +  v). 
We  may  therefore  write 

f[<t>(u  +  v),    <j>'(u  +  v)]  =  0. 

Further,  since  by  hypothesis  <f>(u)  has  an  algebraic  addition-theorem, 
there  exists  an  equation  of  the  form 


As  (/>(v)  is  a  constant,  we  may  determine  (j>(u  +  v)  as  an  algebraic  function 
of  </>(u)  from  this  equation.  It  is  thus  shown  that  the  general  integral 
of  the  differential  equation 


</>'(u)]  =  0 
*  See  Daniels,  loc.  cit.,  p.  256. 


44  THEORY   OF   ELLIPTIC    FUNCTIONS. 

is  an  algebraic  function  of  the  particular  solution  <j>(u).  We  note  that 
this  theorem  is  not  true  for  every  differential  equation  in  which  the  argu 
ment  does  not  enter  explicitly,  but  only  for  those  functions  for  which  there 
exists  an  algebraic  addition-theorem. 

If  one  succeeds  in  integrating  the  differential  equation  in  two  ways, 
the  one  being  by  the  addition  of  a  constant  to  the  argument  of  the  function 
and  the  second  in  any  other  way,  the  addition-theorem  is  at  once  deduced 
by  equating  the  two  integrals.  (See  Chapter  XVI.) 

THE  DISCUSSION  RESTRICTED  TO  ONE-VALUED  FUNCTIONS. 

ART.  40.  We  proceed  next  with  the  consideration  of  the  two  equations 
of  Art.  34: 


7  >  ' 

dv 

,  0  =  0.  (2) 

The  first  of  these  equations  may  be  written  in  the  form 

-  •  +  Ak-i  C  +  Ak  =  0, 


where  the  A's  are  integral  functions  of  £,  77,  £',  T/,  while  the  second  equation 
has  the  form 

+   '     '     *   +  Om-lC  +  am  =  0, 


the  a's  being  integral  functions  of  £,  77. 

By  the  application  of  Euler's  method  for  finding  the  Greatest  Common 
Divisor  of  these  functions,  it  is  seen  that  this  divisor  is  an  integral  function 
of  the  A's  and  a's  and  £,  say 

go:,  f,  r,,  r,  v>-  (3> 

This  function  equated  to  zero  is  the  simplest  equation  in  virtue  of  which 
equations  (1)  and  (2)  are  true.  If  g  is  to  be  a  one-valued  function  of  its 
arguments  and  if  £,  r),  £',  tf  have  each  a  definite  value  for  a  definite  value 
of  u,  then  £  also  must  have  a  definite  value,  so  that  the  equation  (3)  must 
be  of  the  first  degree  in  £.  Hence  £  must  have  the  form 


=  F(?    d£. 
V     du' 


dv 


where  F  is  a  rational  function  of  its  arguments. 

We  shall  leave  for  a  later  discussion  (Chapter  XXI)  the  determination 
of  all  analytic  functions  which  have  algebraic  addition-theorems.  At 
present  we  shall  only  seek  among  such  functions  those  which  have  the 
property  that  £  =  cf>(u  +  v)  may  be  expressed  rationally  in  terms  of 
<j)(u),  <j>'(u),  <l>(v),  fi(v).  All  these  functions  have  the  property  of  being 


ALGEBRAIC   ADDITION  THEOREMS.  45 

one-valued  analytic  functions  of  the  independent  variable.  The  reciprocal 
theorem  is  also  true:  All  analytic  functions  for  which  there  exists  an  algebraic 
addition-theorem  and  which  at  the  same  time  are  one-valued  functions  of  the 
independent  variable,  have  the  property  that  <J>(u  +  v)  may  be  expressed 
rationally  through  <fi(u),  <j>'(u),  (f>(v),  (f>'(v).  Much  emphasis  is  put  upon  this 
theorem,  which  is  proved  in  Art.  158. 

Thus  while  the  general  problem  has  been  restricted,  we  have  in  fact 
only  limited  the  discussion  in  that  one-valued  analytic  functions  are 
treated. 

It  may  be  remarked  here  that  the  rationality  of  <j>(u  +  v)  in  terms  of 
<j)(u),  </>'(u),  (j>(v),  (f>'(v)  is  not  characteristic  of  all  analytic  functions  with 
algebraic  addition-theorems,  but  only  of  one-valued  analytic  functions. 
To  such  functions  for  example  the  remarks  of  Prof.  Forsyth  at  the  con 
clusion  of  Chapter  XIII  of  his  Theory  of  Functions  must  be  restricted.* 

ART.  41.     We  shall  show  (cf.  Schwarz,  loc.  ciL,  p.  2)  that 

I.  All  rational  functions  of  the  argument  u,  and 

um 

II.  All  rational  functions  of  an  exponential  function  e  w  ,  where  a)  is 
different  from  zero  or  infinity,  have  algebraic  addition-theorems  and  have 
the  property  that  </>(u  +  v)  may  be  expressed  rationally  in  terms  of  <£(w), 


These  functions  are  (cf.  Art.  293)  limiting  cases  of  elliptic  functions; 
those  under  heading  I  are  not  periodic  and  those  under  II  are  simply 
periodic.  Finally,  we  have 

III.  The  elliptic  functions,  which  are  doubly  periodic.  These  functions 
have  the  properties  just  mentioned  under  I  and  II. 

We  shall  see  in  Art.  78  that  there  do  not  exist  one-valued  functions 
which  have  more  than  two  periods.  Hence  every  function  for  which  there 
exists  an  algebraic  addition-theorem  is  an  elliptic  function  or  a  limiting 
case  of  one. 

ART.  42.     Let  (f>(u)  be  a  rational  function  of  finite  degree  and  let 


By  means  of  these  three  equations  we  may  eliminate  u  and  v  and  then 
have  an  equation  of  the  form 

(A)  (?(£,  i,  C)  =  0, 

where  G  denotes  an  integral  function  of  its  arguments. 
Writing 

(1)     £  =  £(«),  (2)     £-p(u), 

du 

*  Cf.  also  Biermann,  Theorie  der  analytischen  Funktionen,  p.  393,  and  Phragmen, 
Act.  Math.,  Bd.  7,  p.  33. 


46  THEORY   OF   ELLIPTIC    FUNCTIONS. 

we  note  that  both  of  these  expressions  are  algebraic  in  u,  and  by  the  elim 
ination  of  u  we  have  the  eliminant  equation 


(B)  4 1)  •  °- 


which  is  an  ordinary  differential  equation  in  which  the  variable  u  does  not 
appear  explicitly. 

The  equation  (A)  and  the  latent  test  (B)  are  sufficient  to  show  that 
every  rational  function  has  an  algebraic  addition-theorem. 

We  shall  next  show  that  in  the  case  of  the  rational  functions  the  argu 

ment  u  may  be  expressed  rationally  in  terms  of  £  and  —  • 

du 
We  assume  first  that  the  two  equations 

£=0(u)    and     ^-=<t>'(u) 
du 

have  only  one  common  root,  which  may  be  a  multiple  root.  By  the 
method  of  Art.  40  we  derive  an  equation  which  is  either  of  the  first  degree 
in  u,  in  which  case  we  may  solve  with  respect  to  u  and  thus  have  u  ration 

ally  expressed  through  £  and  —  ;  or  it  is  of  a  higher  degree  in  u,  of  the 

du 

form,  say 

a0um  +  aium~l  +  a2um~2  +  •  •   •  +  am  =  0, 


where  the  a's  are  functions  of  £  and  —  • 

du 

Since  this  equation  must  represent  the  multiple  root,  it  must  be  of  the 
form 

a0(u  -  uo)m  =  0. 

This  expression  developed  by  the  Binomial  Theorem  becomes 


It  follows  from  the  theory  of  indeterminate  coefficients  that 

a\ 
or     UQ  =  —  —  —  • 


Since  a\  and  a$  are  integral  functions  of  £  and  —  .  it  is  seen  that  UQ 

du 

may  be  expressed  rationally  through  these  quantities.     We  may  there 
fore  write 


where  R  denotes  a  rational  function. 

We  thus  see  that  for  the  case  where  the  equations 

$  =  <j>(u)     and     *(%-  =  <t>'(u) 
du 


ALGEBRAIC    ADDITION-THEOREMS.  47 

have  only  one  common  root,  we  have 


Further,  since  (/>  and  R  both  denote  rational  functions,  it  is  seen  that 


where  F  denotes  a  rational  function. 

ART.  43.     We  shall  next  show  that  the  two  equations 

,     |^-  =  *'(*) 
du 

cannot  have  more  than  one  common  root.     For  assume  that  they  have 
the  common  roots  HI  and  u2. 
It  follows  that 

(1) 

(2) 
au 

/7— 

Since  these  two  expressions  exist  for  continuous  values  of  c  and  -^ 

du 

we  may  regard  u\  and  u2  as  two  variable  quantities. 
Taking  the  differential  of  (1)  it  follows  that 


If  we  exclude  as  singular  all  values  of  u  for  which 

0'(wi)  =  0  =  0'(w2), 
then  owing  to  the  relation  (2)  we  have 

du\  =  du2, 
or,  ui  =  u2  +  C, 

where  C  is  a  constant. 

If  therefore  the  two  equations  have  two  common  roots,  these  roots  can 
differ  only  by  a  constant. 

We  thus  have 

C). 


This  expression  is  true  for  an  arbitrarily  large  number  of  values  of  MI, 
and  since  the  degree  of  (j)(u)  is  finite  we  must  have  the  identical  relation 

C). 


Further,  for  MI  we  may  write  any  arbitrary  value  in  the  identity,  say 
u\  +  C,  and  we  thus  have 

<£(MI  +  C)  =  ^(MI  +  2  C)  =s 


48  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Hence  the  roots  of  the  identity  are  wx,  m  +  C,  m  +  2  C,  •  •  •  .  If 
then  C  ^  0,  the  equation 

£  =  4>(u) 

has  an  infinite  number  of  solutions.  This,  however,  is  not  true,  since  the 
equation  is  of  finite  degree.  If  follows  that  the  constant  C  =  0  and  conse 
quently  the  two  equations  can  have  only  one  common  root. 

We  have  thus  shown  that  every  rational  function  of  the  argument  u  has 
an  algebraic  addition-theorem  and  has  the  property  that  <j>(u  +  v)  may  be 
rationally  expressed  through  (/>(u),  <j>'(u),  <j>(v),  <f>'(v). 

ART.  44.  We  shall  next  show  that  the  theorem  of  the  last  article  is 
also  true  for  all  functions  that  are  composed  rationally  of  the  exponen- 

tun 

tial  function  e  "  . 

Let  /*  be  a  real  or  complex  quantity  different  from  0  and  oo  and  write 

t  =  e^u,  and 
^(0  -  0(w),  (1) 

where  ^  denotes  a  rational  function. 
Further,  let 

s  =  e^v  and 

1?(s)  =<t>(v).  (2) 

It  follows  that 


W  -  s)  =  <t>(u  +  v).  (3) 

From  the  three  equations  (1),  (2)  and  (3)  we  may  eliminate  £  and  y, 
and  have 

(A)  G{<t>(u),<t>(v),  $(u  +  v)}  =0, 

where  G  denotes  an  integral  function. 

We  have  under  consideration  a  group  of  one-  valued  analytic  functions 
which  have  everywhere  the  character  of  an  integral  or  fractional  func 
tion  and  which  are  simply  periodic,  the  period  of  the  argument  being 

^  =  2  co,  say. 
P 
We  further  have 


If  t  is  eliminated  from  these  equations,  we  have  the  eliminant  equation 
(B)  '  /(*,£)-  0, 

where  /  denotes  an  integral  function. 


ALGEBRAIC   ADDITION-THEOREMS.  49 

It  follows  from  equations  (A)  and  (B)  that  the  function  <j>(u)  has  an 
algebraic  addition-theorem. 

ART.  45.  It  may  be  shown  as  in  the  case  of  the  rational  functions 
that  when  the  equations 

£  =  V(0     and     ^-  =  ^(t)fit 
du 

have  one  common  root  *  in  t,  then  we  may  express  t  in  the  form 


duj 

where  R  denotes  a  rational  function.     It  also  follows  that 
^(u  +  v)  =  F[</>(u),  P(u),  <}>(v),  <t>'(v)], 

where  F  is  a  rational  function. 

Suppose  next  that  the  two  equations 


have  more  than  one  common  root. 

Suppose  that  t\  and  t2  are  two  roots  that  are  common  to  both  equa 
tions,  so  that 

(1) 


(2) 
du 

If  we  consider  ^i  as  the  independent  variable,  then  t2  is  an  algebraic 
function  of  t\,  since 


and  ^  is  a  rational  function. 

From  equation  (1)  it  follows  that 


which  divided  by  the  expression  (2)  becomes 

dt±  =  dtz 
ti   "  t2  ' 

or  log  ti  =  log  ^2  —  log  C,     so  that 

t2  =  Ch. 

It  is  thus  seen  that  if  the  two  given  equations  have  two  common  roots, 
these  roots  can  differ  only  by  a  multiplicative  constant.     Since 

^(tz),  it  follows  that 


which  is  an  algebraic  equation  of  finite  degree. 

*  By  equating  the  discriminant  to  zero,  we  may  always  effect  the  condition  that 
there  is  one  common  root. 


50  THEORY   OF   ELLIPTIC   FUNCTIONS. 

As  this  equation  can  be  satisfied  by  an  infinite  number  of  values  of 
it  must  be  an  identical  equation  and  consequently 


It  follows  at  once  that 


But  the  equation  yfr(ti)=TJr  (t2)  being  of  finite  degree  with  respect  to  t2 
can  only  be  satisfied  by  a  finite  number  of  different  values  of  t2.  It 
therefore  follows  that  in  the  series  of  quantities 

Ctlf  C2*i,  C%,  .  .  .  ,  CP*I,  .  .  .  ,  C%,  .  .  .  ,  (1) 

some  must  have  equal  values.  If  the  degree  of  the  equation  is  n  in  t2 
then  among  the  first  n  +  1  of  these  quantities  two  must  be  equal,  say 

CP  =  C«         (q>  p). 
Writing  q  —  p  =  ra,  a  positive  integer,  we  have 

Cm  =  1.  (2) 

It  is  thus  shown  that  C  is  an  mth  root  of  unity,  and  as  m  is  the  smallest 
integer  that  satisfies  this  equation  it  is  a  primitive  mth  root  of  unity.  It  is 
easy  to  see  that  the  quantities 

C,  C2,  C3,  .  .  .,  Cm~\Cm 

are  all  different.     For  if 

&-&(i,3£m), 

then  is  C*'~J'  =  1  (where  i  —  j  =  m'  <m). 

This,  however,  contradicts  the  hypothesis  that  m  is  the  smallest  integer 
which  satisfies  the  equation  (2).  There  are  consequently  only  m  different 
quantities  in  the  series  (1). 

We  may  use  this  fact  and  employ  the  identical  equation 


to  show  that  the  rational  function  ^r(Ji)  may  be  transformed  into  another 
rational  function  ty\(tm).  If  then  we  write  r  =  tm,  we  may  substitute 
the  function  ^i(r)  in  the  above  investigation  in  the  place  of  ty(t),  where 

the  degree  of  the  equation  in  T  is  —  ,  n  being  the  degree  of  the  equation  in  t. 

m 

The  function  ^r(t)  may  be  expressed  as  the  quotient  of  two  integral 
functions  without  common  divisor  in  the  form 

t±»  A  (t-ai)(t-a2)  .  .  .   (t  -a,) 
•    (t  -  &!)  (t  -  62)  .  .  .   (t  -  W  ' 

where  //  is  an  integer  and  where  none  of  the  a's  or  6's  is  equal  to  zero. 


ALGEBRAIC   ADDITION-THEOREMS.  51 

Further,  since  ^(t)  =  ^(Ct),  we  must  have 

\  f  +  u    1     ft"    fll)     (*    -    ^     •     •     •      (*    -    a") 

(t  -  60  (*  -  62)  .  . 

*  -  QI)  (Ct  -  o2)  . 


(C/  -  &0  (C*  -  62)  .  .  .   (Ct  -  bff) 

The  left-hand  side  of  this  equation  is  zero  for  t  =  a\;  it  follows  that  the 
right-hand  side  must  also  vanish  for  this  value  of  t.  But  Cai  —  0,1  ^  0, 
if  we  assume  that  C  5^  1.  Hence  one  of  the  other  factors  must  be  such 
that  Cai  —  a  A  =  0,  where  A  is  to  be  found  among  the  integers  2,  3,  .  .  .  ,  p. 
As  it  is  only  a  matter  of  notation,  we  may  write  A  =  2,  so  that 

Cai  -  a2  =  0,     or     a2  =  Cax. 

In  a  similar  manner,  since  the  left-hand  member  of  the  equation  van 
ishes  for  a2,  one  of  the  factors  on  the  right-hand  side  must  vanish  for 
t  =  a2,  say  Ca2  —  av  =  0,  where  v  is  to  be  found  among  the  integers 
1,  3,  4,  .  .  .  ,  p,  say  v  =  3. 

We  thus  have 

Ca2  —  a3  =  0,     or     a3  =  Ca2  =  C2ai. 
Continuing  this  process  we  derive  the  relations 

a>i  =  a.i,  a2  =  Cai,  a3  =  C2ab  .  .  .  ,  am  =  Cm-ldi. 
Further,  since  C,  C2,  .  .  .  ,  Cm~l  are  all  different,  it  is  seen  that 

01,  a2,  .  .  .  ,  am 
are  all  different. 

The  quantities  ab  Cai,  C2a1?  .  .  .  ,  Cm~1ai  form  a  group  of  roots  of 
the  equation,  and  after  Cote's  Theorem 


(t  -  ai)  (t  -  Cai)  (t  -  C2a!)  .  .  .   (t  -  Cm~lal)  =  tm  -  a^. 

This  factor  tm  —  aim  may  consequently  be  separated  from  the  two  sides 
of  the  equation  (I).  If  further  there  remain  linear  factors  in  the  numer 
ator  of  equation  (I),  we  repeat  the  above  process  until  there  are  no  such 
factors.  The  same  is  also  done  with  the  denominator.  When  all  such 
factors  have  been  divided  out  from  either  side  of  the  equation  (I),  there 
remains 


so  that  Cft  =  1.     It  follows  at  once  that  jj.  must  be  a  multiple  of  m  and 
consequently 


A  (t 


±—  (tin 

m  (     ~ 


(tm  -  6^)  (tm  - 


52  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  have  thus  shown  that  if  the  two  equations 

?  =  *(«),  I1  =  <£'(«) 

du 

have  more  than  one  root  in  common,  there  exists  an  integer  m,  such 
that  fat)  may  be  expressed  as  a  rational  function  of  tm.  • 

Writing  t  =  e»u,  it  follows  that  tm  =  em(iu   and 


In  the  further  discussion  we  may  use  ty\(tm)  in  the  place  of 
It  may  happen  that  the  two  equations 

£  =  ^i(emftu)    and    —  =  fa  (em^u)  m  u.emtiU 
du 

have  more  than  one  common  root.  By  repeating  the  above  process  we 
may  diminish  the  degree  of  ^i  and  replace  the  function  ^1(em'iU)  by  the 
equivalent  function  fa(emm'ftu)J  where  m'  is  an  integer,  etc. 

Since  the  original  function  -^  was  of  finite  degree,  a  finite  number  of 
divisors  must  reduce  the  degree  to  unity.  It  therefore  follows  that  in  the 
process  of  diminishing  the  degrees  of  the  functions  -^,  fa,  fa,  .  .  .  ,  we  must 

come  to  a  function,  say  £  =  iK;  such  that  £  and  —  have  no  common 

du 

root  for  the  new  variable  that  has  been  substituted  in  fa.  Hence  with 
out  exception  the  following  theorem  is  true: 

I.    All  rational  functions  of  the  argument  u;  and         u^i 
II.    All  rational  functions  of  the  exponential  function  e  u    have  algebraic 
addition-theorems  and  are  such  that 


<j>(u  +  v)  = 
where  F  denotes  a  rational  function. 

Example.  —  Apply  the  above  theory  to  the  examples  sin  u,  cos  u,  tan  u.     Write 

piu  _  p  -  iu  i      /2  _  1 

smu  =  -  -  -  -  =  —  "  -  -.    where  t  =  elu. 
2i  2  1      t 

ART.  46.  It  may  be  shown  by  an  example  that  a  function  <j>(u)  may 
have  the  property  that  <p(u  +  v)  is  rationally  expressible  through  (/>(u), 
<f>'(u),  <f>(v),  <t>'(o)  without  having  an  algebraic  addition-theorem. 

Take  the  function 

cj)(u)  =  Aeau  +  Bebu,  (1) 

where  A,  B,  a,  b  are  constants  and  a  ^  b.     It  follows  that 

<t>'(u)  -  aAeau  +  bBebu.  (2) 


ALGEBRAIC    ADDITIOX-THEOKEMS.  53 

From  (1)  and  (2)  we  have 


b  -  a 


b  -  a 
We  further  have 

u  +  v)=  Aeaueav  +  Bebuebv 


A          b  —  a  b  —  a  B  b  —a  b  —  a 

from  which  it  is  seen  that  (j>(u  +  v)  may  be  expressed  rationally  in  terms 
of  0(M),  0'(u),  0(v),  0'(t>). 

We  shall  now  show  that  0(u)  has  no2  an  algebraic  addition-theorem. 

We  so  choose  a  and  b  that  the  ratio  -  is  an  irrational  or  complex  quan- 
tity. 

In  Art.  35  we  saw  that  without  exception  the  differential  equation 


where  /  denoted  an  integral  algebraic  function,  existed  for  all  functions 
which  had  algebraic  addition-theorems.  If  therefore  we  can  prove  that 
such  an  equation  does  not  exist  for  <j>(u},  we  may  infer  that  (f>(u)  does  not 
have  an  algebraic  addition-theorem. 

Suppose  for  the  function  <f>(u)  there  exists  an  equation  of  the  form 


where  /  denotes  an  integral  function. 

Since  (j>(u)  and  (f>'(u)  may  be  expressed  through  eau  and  ebu  where  only 
constant  terms  occur  in  the  coefficients,  we  may  write  the  above  equa 
tion  in  the  form 

fi[eau,  ebu], 

where  /i  like  /denotes  an  integral  function  of  finite  degree.     This  equation 
must  be  satisfied  for  all  values  of  u  for  which  the  function  (f>(u)  is  defined. 
We  give  to  u  successively  the  values 

.   2/ri  .   4:7n 

U0,    U0   H  --  ,     UQ  -\  --  1    •    '    •      . 

a  a 

The  quantity  eau  has  the  same  value,  viz.,  eau«  for  all  these  values  of  u. 
But  corresponding  to  one  value  of  eaUfl,  the  equation  above  being  of  finite 
degree  can  furnish  only  a  finite  number  of  different  values  of  ebu.  On 


54 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


the  other  hand  there  correspond  to  the  one  value  eau°  an  infinite  number 
of  values  ebu  of  the  form 


ebu°,    e 


bu0+-2ni 


b  . 


which  are  all  different,  since  the  ratio  -  is  not  rational. 

a        /     d£\ 

It  follows  that  the  eliminant  equation  f\£.  —  )  =  0  does  not  exist  for 

V    duj 

the  given  function,  and  consequently  this  function  does  not  have  an  alge 
braic  addition-theorem.  We  have  thus  proved  that  the  existence  of  the 
relation 


F  denoting  a  rational  function,  does  not   necessarily  imply  the  existence 
of  an  algebraic  addition-theorem. 


CONTINUATION  OF  THE  DOMAIN  IN  WHICH  THE  ANALYTIC  FUNCTION 

HAS  BEEN  DEFINED,  WITH  PROOFS  THAT  ITS  CHARACTERISTIC  PROP 
ERTIES  ARE  RETAINED  IN  THE  EXTENDED  DOMAIN. 

ART.  47.  In  the  previous  discussion  we  have  supposed  that  </>(u)  was 
denned  for  a  certain  region  which  contained  the  origin.  This  region  we 
may  call  the  initial  domain  of  the  function  </>(u).  We  further  assume 
that  <f>(u)  has  an  algebraic  addition-theorem  and  is  such  that  </>(u  +  v) 
may  be  rationally  expressed  through  (f>(u),  <£'(^),  <£M,  0'M  within  this 
initial  domain. 

These  properties  are  expressed  through  the  two  equations 


(II)  <j>(u  +  v)=F{<f>(u),  <f>'(u),  </>(v),  P(v)}, 

where  G  denotes  an  algebraic  function  and  F  a  rational  function. 

We  also  assume  that  u  and  v  are  taken  so  that  u  +  v  lies  within  the 
initial  domain.* 

We  shall  now  prove  the  following  theo 
rem:  If  the  function  <j)(u)  has  the  properties 
above  mentioned,  it  has  the  character  of  an 
integral  or  a  (fractional)  rational  function 
in  the  neighborhood  of  the  origin. 
In  the  equation  (II)  we  write 
u  +  v  in  the  place  of  u, 
—  v  in  the  place  of  v, 

u  in  the  place  of  u  +  v. 
We  thus  have 

-  v),  $'(-  v)  }  .  (1) 


Fig.  3. 


<t>(u)'  =  F\<i>(u  +  v),  <f>'(u  +  v), 
*  Cf.  Weierstrass,  Abel'schen  Functionen,  Werke  4,  pp.  450  et  seq. 


ALGEBRAIC   ADDITION-THEOREMS.  55 

Such  values  are  chosen  for  v  that  for  these  values  the  functions  (f>(v) 
and  (f)(—  v)  belong  to  the  initial  domain.  We  develop  (f>(u  +  v)  by 
Taylor's  Theorem  in  the  form 

<j>(u  +  v)  =  <f>(v)  +  u0'(t>)  +  ~0»  +  •  •  •  .' 

a  series  which  remains  convergent  so  long  as  v  takes  such  values  that  the 
points  u  +  v  and  u  lie  within  the  initial  domain.  The  same  is  also  true 
of  the  series 

4>'(u  +  v)  =  P(v)  +  u4>"(v)  +  |J0"'M  +  •••.• 

These  series  may  therefore  be  substituted  in  formula  (1).  We  thus  have 
(/)(u)  expressed  as  a  rational  function  of  u,  which  as  the  quotient  of  two 
integral  functions  takes  the  form 

QO  +  a  iu  +  a2u2  +  •   •   • 


where  the  two  series  are  convergent  so  long  as  |  u  |  is  less  than  a  certain 
quantity,  say  p. 

If  60  T^  0,  <!>(u)  has  the  character  of  an  integral  function  in  the  neigh 
borhood  of  the  origin  u  =  0;  if  60  =  0  =  61  =  •  •  •  =  bk 

and  at  the  same  time 

a0  =  0  =  ai  =  •  •  •  =  a*, 

then  (f>(u)  has  the  character  of  an  integral  function  at  the  origin;  but  if 
one  of  the  a's  just  written  is  different  from  zero,  then  <f>(u)  becomes  infinite 
for  u  =  0  but  of  a  finite  integral  degree.  It  then  has  the  character  of  a 
rational  function  at  the  origin,  and  its  expansion  by  Laurent's  Theorem* 
has  a  finite  number  of  terms  with  negative  integral  exponents. 

ART.  48.  We  may  next  prove  the  following  theorem:  The  domain  of 
tf>(u)  may  be  extended  to  all  finite  values  of  the  argument  u  without  the  func 
tion  (j)(u)  ceasing  to  have  the  character  of  an  integral  or  (fractional)  rational 
function. 

Fundamental  in  the  proof  of  this  theorem  is  the  expression  of  (f>(u)  as 
the  quotient  of  two  power  series 


where  the  two  series  are  convergent  so  long  as  |  u  |  does  not  exceed  a  definite 
limit  p. 

If  we  draw  the  circle  with  radius  p  about  the  point  u  =  0,  then  within 

*  In  this  connection  see  a  proof  of  Laurent's  Theorem  by  Professor  Mittag-Leffler, 
Acta  Math.,  Bd.  IV,  pp.  80  et  seq.,  where  the  theorem  is  proved  by  the  elements  of  the 
Theory  of  Functions  without  recourse  to  definite  integrals. 


56  THEORY    OF   ELLIPTIC    FUNCTIONS. 

this  circle  the  function  <j>(u)  is  completely  denned.     In  order  to  extend  or 
continue  this  region,  we  may  use  the  equation 


<t>(u  +  v)  = 

We  shall  at  first  assume  that  we  may  write  u  =  v  without  the  function  F 
taking  the  form  0/0.     We  then  have  *  for  u  =  v, 

u)  = 


The  right-hand  side  of  this  equation  is  true  for  all  values  of  u  that  lie 
within  the  circle  with  radius  p.  It  follows  then  that  through  this  expres 
sion  the  function  <j>  on  the  left-hand  side  is  defined  so  long  as  its  argument 
lies  within  the  circle  with  radius  2  p. 

If  then  we  write  u  in  the  place  of  2  u  in  this  equation,  we  have 

<j>(u)  =  F  {</>($  u\<t>'Qu),<t>Q  *,),<!>'  (I  u)}. 

We  express  <j>(u),  as  the  quotient  of  two  power  series,  =     1>0^  .  Further, 

™i»oOO 
<f>'(u)  may  also  be  expressed  as  the  quotient  of  two  power  series.     These 

values  substituted  in  F  give  (f>(u)  defined  as  the  quotient  of  two  new  power 
series,  say 


Since  \  u  has  been  written  for  u  in  the  two  new  power  series,  they  are 
convergent  so  long  as  |  u  <  2  p. 

We  cannot  apply  the  above  method,  if  for  u  =  v  the  function  </>(u) 
takes  the  form  0/0.  Nevertheless  we  may  proceed  as  follows  and  extend 
the  region  of  convergence  at  pleasure. 

In  the  equation 

#u  +  v)  =  F\4>(u),  #(u),  <t>(v),  4/(v)}, 

we  write  —  -  —  instead  of  u, 

1  +  a 

and  au    instead  of  v, 

1  +a 

where  a  is  a  real  quantity  such  that  J  <  a  <  1. 
We  have  in  this  manner 


The  function  F  being  a  rational  function,  we  may  express  (f)(u)  as  the 
quotient  of  two  power  series  in  which  the  numerator  and  denominator  are 

*  See  Daniels,  Amer.  Journ,  Math.,  Vol.  VI,  p.  255. 


ALGEBRAIC    ADDITION-THEOREMS.  57 

analytic  functions  of  u  and  a.  The  denominator  cannot  vanish  for  all 
values  of  a.  We  shall  therefore  so  choose  a  that  the  denominator  is  differ 
ent  from  zero.  We  may  then  express  <j>(u)  as  the  quotient  of  two  power 
series  in  the  form 


where  the  series  are  convergent  for  values  of  u  such  that 

|  u  |  <  (1  +  a) p. 

Since  a  =  J  the  series  is  convergent  if  |  u  \  <  f  p. 

This  process,  as  well  as   the   one  employed   in  the  previous   article, 
may  be  repeated  as  often  as  we  wish,  so  that  we  have  eventually 

P2,n(u)' 
where  the  power  series  Pi§n(u)  and  P2,n(u)  are  convergent  so  long  as 

Hence  (j)(u)  may  be  defined  for  an  arbitrarily  large  portion  of  the  plane  as 
the  quotient  of  two  power  series  which  may  be  expanded  in  ascending 
powers  of  u.* 

ART.  49.     As  an  example  of  the  above  theory,  consider  the  function    ' 


cos  u      P2u  3         15 


where  Plt0(u)  =  u  -       4-       - 


At  the  points  0,  n,  2  ^,  3  it,  .  .  .  ,  the  function  tan  u  is  zero,  and  is 
infinite  at  -,  —  ,  —  ,  •  •  •  . 

For  the  point  u  =  °c  ,  the  function  tan  u  is  not  defined,  this  point  being 
an  essential  singularity  of  the  function.  The  function  is  convergent  for 
all  points  within  a  circle  described  about  the  point  u  =  0,  whose  radius 

extends  up  to  the  infinity  f1  of  tan  u,  so  that  we  may  take  p  =  '-  . 

*  Weierstrass,  Werke  IV,  p.  6,  says  that  from  the  fact  that  <£(u)  has  an  algebraic 
addition-theorem  we  may  show  that  it  is  a  uniquely  denned  function  having  the  char 
acter  of  an  integral  or  rational  (fractional)  function  and  that  starting  with  this  we 
may  derive  a  comul^te  theory  of  the  elliptic  functions. 


58  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Using  the  formula 

tan  («  +  „)        tanit  +  tanr. 


1  —  tan  u  tan  v 

we  may  extend  the  definition  of  tan  u  to  an  arbitrarily  large  region.     For 
writing  v  =  u,  then  is 

tan2«=     2tan« 


1  —  tan2  u 
Further,  if  we  put  ^  u  in  the  place  of  u,  we  have 


1  -  tan2  i  u      1  -  P2(£  u)       P2,  i  (w)  ' 
where  PI,  i(u)  and  P2,  \(u)  are  convergent  so  long  as 

i  |  M  |  <  i  TT     or     |^|<TT. 

We  see  that  here  the  new  circle  of  convergence  passes  through  the  points 
+  TT  and  —  TT  and  that  the  old  region  of  convergence  has  been  extended 
by  a  ring-formed  region. 

By  another  repetition  of  the  same  process  we  have 


tana  =  ^^       _^l,2i«) 

1       

The  radius  of  convergence  of  the  two  series  on  the  right-hand  side  is  now 
2  TT,  so  that  the  tangent  function  is  defined  for  all  points  within  the  circle 
whose  radius  is  2  n.  By  continuing  this  process  we  are  able  to  define  tan  u 
for  all  finite  values  of  the  argument  u  without  its  ceasing  to  have  the 
character  of  an  integral  or  (fractional)  rational  function. 

ART.  50.  Returning  to  the  general  case  we  shall  see  whether  the  function 
which  has  been  thus  defined  for  all  points  of  the  plane  is  the  same  as  the 
function  (f>(u)  with  which  we  started  and  which  was  defined  for  the  interior 
of  the  circle  with  the  radius  p.  We  shall  show  that  such  is  the  case  and 
that  the  new  function  is  the  analytic  continuation  of  the  one  with  which 
we  began.*  We  shall  first  show  that  the  two  functions  are  identical 
within  the  circle  with  radius  p. 

It  is  seen  that  the  expression  of  <j>(u)  as  the  quotient  of  two  convergent 
power  series  is  characteristic  of  this  sort  of  function.  We  limit  u  to  the 
interior  of  the  circle  with  radius  j-  p  within  which  v  is  also  restricted  to 
remain.  The  points  u,v,u  +  v  evidently  lie  within  the  domain  for  which 
<t>(u)  was  defined,  and  the  property  expressed  through  the  formula 

is  true  for  this  domain. 

*  Weierstrass  (Definition  der  Abel'schen  Functionen,  Werke  4,  pp.  441  et  seq.)  empha 
sizes  this  fact. 


ALGEBRAIC   ADDITION-THEOREMS.  59 

On  the  right-hand  side  we  again  write  i 

-  instead  of  u 

and  au     instead  of  v, 

1  +  a 

with  the  limitation  that  the  absolute  values  of  these  quantities  be  less 
than  J  p. 

T}  /  -  .\ 

Writing  first  (f>(u)  =      1>0^  ''    and  then  making  the  formal  computation 

as  above,  we  have  $(u)  =     ltl        •    These  two  quotients  are  identical* 
within  the  circle  with  radius  ^  p,  so  that. 


or 

(1) 

If  we  multiply  these  two  power  series  on  either  side  of  the  equation,  we 
will  have  the  equality  of  two  new  power  series,  which  is  true  for  all  values 
of  u,  such  that  |  u  |  <  J  p.  Now  P1>0  and  PI,  i  are  convergent  by  hypothe 
sis  within  the  circle  of  radius  p,  while  P2,o  and  ^2,1  are  convergent  within 
the  circle  of  radius  f  p  .  Within  the  circle  with  radius  J  p  the  coefficients 
of  u  on  either  side  of  the  equation  are  equal.  But  as  these  coefficients  are 
constants  we  conclude  that  the  two  series  on  the  right  and  left  of  equation 
(1)  must  be  the  same  within  the  extended  realm,  the  circle  with  radius  p. 
It  follows  that  the  representations  of  <f>(u)  through  the  two  quotients 

Pl'°^  and  Pl-1^  are  the  same  within  the  interior  of  the  circle  p.     The 

P2,oO)  P&i'OO 

same  process  may  be  continued  so  as  to  extend  over  the  whole  region  of 

convergence. 

ART.  51.  We  shall  next  prove  that  as  the  definition  of  the  function 
$(u)  is  extended  to  an  arbitrarily  large  region,  the  properties  of  the  original 
function  <£(M)  that  are  expressed  through  the  equations, 


(I)  G{<j>(u),<!>(v),<l>(u  +  v)}  =0, 

(II) 


are  also  retained  for  the  extended  region,  f     First  take  |  u  |  <  J-  p  and 
|  v  |  <  \  p  so  that  |  u  +  v  |  <  p  and  therefore  lies  within  the  initial  domain. 

*  Weierstrass,  loc.  cit.,  p.  455. 

t  This  theorem  has  the  same  significance  for  the  properties  of  the  elliptic  functions 
as  the  fact  that  the  functions  themselves  may  be  analytically  continued  as  emphasized 
in  Chapter  I. 


60  THEORY   OF   ELLIPTIC   FUNCTIONS. 

In  the  equation  G  =  0  write  ^>M  for  $(u)  ^LoM  for  ^  and  PI,O(U  +  V) 


for  (f>(u  +  v).  Multiply  the  expression  thus  obtained  by  the  least  com 
mon  multiple  of  the  denominators  and  we  have  an  integral  power  series  in 
u  and  v  on  the  left  equated  to  zero.  This  power  series  is  convergent 
so  long  as  \u\  <  \  p  and  v  \  <  \  p.  If  this  power  series  is  arranged  in 
ascending  powers  of  u,  the  coefficients  are  functions  of  v  which  may  also 
be  arranged  in  ascending  powers  of  v.  Since  the  right-hand  side  is  zero, 
the  coefficients  of  u  are  all  zero  and  consequently  the  power  series  in 
v  are  identically  zero.  Making  use  of  equation  (II)  we  derive  the  second 
development  for  (f>(u),  viz., 


This  value  and  the  corresponding  values  of  (£>(v),  (f>(u  +  v)  are  now  sub 
stituted  in  (I).  We  thus  make  another  integral  power  series  in  u  and  v 
on  the  left  equal  to  zero  on  the  right  as  in  the  previous  case. 

These  two  power  series  must  be  the  same  so  long  as  |  u  <  i  p  and 
|  v  |  <  \  p.  But  as  here  the  coefficients  of  u  are  all  identically  zero,  this 
must  also  be  true  in  the  extended  region.  By  repeating  this  process 
we  have  the  theorem: 

The  addition-theorem  while  limited  to  a  ring-formed  region,  exists  for  the 
whole  region  of  convergence  established  for  the  function  <f>(u). 

If  the  point  u  =  QO  is  an  essential  singularity,  the  function  (j>(u)  will 
have  this  point  as  a  limiting  position,  that  is,  the  function  may  be  con 
tinued  analytically  as  near  as  we  wish  to  this  point,  but  at  the  point  QO 
the  function  need  have  the  character  of  neither  an  integral  nor  a  (frac 
tional)  rational  function. 


CHAPTER  III 
THE  EXISTENCE  OF  PERIODIC  FUNCTIONS  IN  GENERAL 

Simply  Periodic  Functions.     The  Eliminant  Equation. 

ARTICLE  52.  In  the  previous  Chapter  we  have  studied  the  characteristic 
properties  of  one-valued  analytic  functions  which  have  algebraic  addition- 
theorems.  These  properties  were  considered  in  the  finite  portion  of  the 
plane.  The  function  may  behave  regularly  at  infinity  or  this  point  may 
be  either  a  polar  or  an  essential  singularity  of  the  function.  In  the  latter 
case  the  function  is  quite  indeterminate  (Art.  3)  in  the  neighborhood  of 
infinity. 

When  the  point  at  infinity  is  an  essential  singularity,  we  shall  show 
that  the  function  is  periodic.  To  prove  this  we  have  only  to  show  that 
the  function  may  take  certain  values  at  an  arbitrarily  large  number  of 
points  (cf.  Art.  38)  of  the  w-plane. 

Suppose  that  m  is  the  number  of  points  at  which  <p(u)  =  £o,  say,  where 
£o  is  a  definite  constant,  and  denote  these  points  by  ai,  a2,  .  .  .  ,  am. 

Let  a,,  be  any  one  of  these  points,  and  with  a  radius  r^  draw  a  circle  0? 
about  0,^  as  center.  Take  r^  so  small  that  within  and  on  the  periphery 
of  Op  none  of  the  other  points  a\t  a2,  .  .  .  ,  c^-i,  aft+i,  .  .  .  ,  an  lies, 
and  also  within  and  on  the  periphery  of  this  circle  suppose  that  (f>  (u  )  is 
everywhere  regular.  Next  let  u  take  a  circuit  around  C  in  the  w-plane; 
then  in  the  plane  in  which  <p(u)  is  geometrically  represented  (f>(u)  makes 
a  closed  curve  S^  say,  which  does  not  pass  through  the  point  £o- 

We  may  write 

_-  -  d\<t>(u)  -  $Q\ 


and  expressing  <j>(u]  —  £o  in  the  form 


it  is  seen  that 


d<fr(u)          d{rei&}      e^dr       rieiedO 
i9  " 


61 


62  THEORY   OF   ELLIPTIC    FUNCTIONS. 

If  next  we  integrate  around  S^  in  the  </>(u)  -plane,  we  have 


The  first  integral  on  the  right  is  log  r,  which  is  here  zero,  since  the  curve 
returns  to  its  initial  point,  making  the  upper  and  lower  limits  identical. 
We  thus  have 

r  J^L.=  r  w. 

JSptpW)  —  <f0       JS* 

On  the  other  hand, 


r    d4>(u)       r   </>'(u)d 

J8fi  </>(u)—  £0       JCft  (f)(u)  — 


where  the  integration  of  the  first  integral  is  taken  with  respect  to  the 
elements  d$(u)  and  consequently  over  S^  in  the  </>(u)  -plane,  while  the 
integration  over  the  second  integral  is  with  respect  to  du  and  there 
fore  over  the  circle  C^  in  the  w-plane.  The  function  <l>(u)  when  developed 
in  powers  of  u  —  a^  is  of  the  form 


or,  snce 


On  the  right-hand  side  a  number  of  the  coefficients  may  vanish.     Let 
AU—  '    j         be  the  first  of  the  coefficients  that  is  different  from  zero  and 

/C  ! 

let  k  =  kp,  say,  be  the  order  of  the  zero  of  the  function  <j>(u)  -  £0  at  the 
point  aft. 

We  therefore  have 


and  consequently 

C      d4>(ii)     __    C  A^k^u  —  g^)*^"1  +  •   •   • 
JSf!(j)(u}  —  £o      Jcft     Afl(u  —  a^)kfi  +  •  -  - 

all  the  remaining  terms  having  vanished. 

or  C     k.,du 

Since  I    — ^ =  2  mk^ 

kfffri  =    I    idd,     or     k»  = I     rf^. 

Jsu  2  rd  Jsv 


it  follows  that 


PERIODIC    FUNCTIONS  IN    GENERAL.  63 

In  other  words,  the  order  of  zero  of  the  function  <j>(u)  —  co  at  the  point 
u  =  dp,  that  is,  kf,  is  equal  to  the  number  of  circuits  which  the  curve 
in  the  (j)(u) -plane  makes  around  co  corresponding  to  the  circle  C ^  made 
around  the  point  a^  by  the  variable  u  in.  the  w-plane.  The  integer  k^  is 
at  least  unity. 

Suppose  in  the  place  of  a^  another  point  a0  is  written,  and  about  this 
point  let  a  circle  Co  be  described  with  a  radius  so  small  that  within  and 
on  the  circumference  of  the  circle  none  of  the  points  01,  a2,  .  .  .  ,  an  lies, 
nor  any  of  the  infinities  of  the  function.  We  know  then  that  the  integral 


I 


—  co 


s  zero, 


where  the  path  of  integration  is  taken  over  the  circle 
We  have  accordingly  proved  *  that  the  integral 


2*»V*#(«)  -to' 

where  $(u)  is  a  regular  function  for  all  points  on  and  within  the  interior  of 
the  contour  S,  indicates  the  number  of  times  that  the  function  <£(u)  takes  the 
value  £o  within  S,  provided  each  point  a^  say,  at  which  <j>(u)  takes  the  value 
£o,  is  counted  as  often  as  the  order  ku  of  the  zero  of  <j>(ii)  —  co  <&  the  point  a^. 

ART.  53.  Next  in  the  place  of  co  take  another  value  ci,  which  also 
lies  within  Sw  so  that  the  corresponding  value  of  u  lies  within  C^ .  Then 
the  number  of  circuits  of  the  curve  about  co  is  the  same  as  the  number  of 
circuits  about  co,  since  all  the  circuits  encircle  both  points. 

It  follows  that 


7    J'    /        \  /"* 

„  <t>(u)  —  ci        '"      Jcf 


<f>(u)  -  c 


and  consequently  that  (j>(u)  takes  for  values  of  u  within  C  \  the  value 
£1  at  least  once;  for  if  this  were  not  the  case,  we  know  that  the  above 
integral  would  vanish.  We  have  shown  above  that  it  does  not  vanish. 

The  function  <j>(u)  by  hypothesis  takes  the  value  £o  on  the  m  different 
points  a  i,  a2,  .  .  .  ,  aw  .  .  .  ,  am.  Around  each  of  these  points  a  circle 
is  drawn  with  radius  sufficiently  small  that  within  the  interior  of  the 
circle  none  of  the  other  points  of  the  series  ab  a2,  .  .  .  ,  am  lies.  Let  u 
make  a  circuit  about  the  periphery  of  each  of  these  circles;  then  <j>(u) 
makes  closed  curves  about  the  point  £o,  and  none  of  these  curves  passes 
through  co-  We  may  therefore  draw  a  circle  about  co  which  lies  within 
all  the  other  closed  curves.  Let  £1  be  a  point  within  the  interior  of  this 
last  circle;  then  it  follows  from  above  that  the  function  </>(it)  takes  the 
value  £1  at  least  m  times.  There  are  consequently  an  infinite  number 

*  See  footnote  to  Art.  92. 


64  THEORY   OF   ELLIPTIC    FUNCTIONS. 

of  values  in  the  neighborhood  of  £o  which  are  taken  by  the  function  at 
least  m  times. 

Consider  next  the  function  -  -- 

<j)(u)  -  £o 

It  has  at  the  point  ai}  a2,  .  .  .  ,  am  the  character  of  a  (fractional) 
rational  function,  and  may  therefore  be  expanded  by  Laurent's  Theorem  in 
the  form  * 


u  - 
+  an  integral  function  in  u, 


-L-) 

\u  -  amj 


where  G\,  G2,  .  .  .  ,  Gm  denote  integral  functions  of  finite  degree  of  their 
respective  arguments. 
It  follows  that 

i  „  /    l    V  „  /    1    S  l-Y  •  1  'S     p(u) 


where  P(u)  is  a  power  series  with  positive  integral  exponents. 

The  function  P(u)  cannot  reduce  to  a  constant,  for  then  </>(u)  would  be 
a  rational  function  and  the  point  u  =  oo  would  not  be  an  essential  singu 
larity.  It  follows  that  the  absolute  value  of  the  above  difference  exceeds 
any  limit  if  we  take  values  of  u  sufficiently  distant  from  the  origin.  We 
may  therefore  by  taking  u  sufficiently  large  make  <j>(u)  —  £o  as  small  as 
we  wish. 

If  further  the  point  £1  is  taken  very  near  the  point  £Q>  the  value  £1  is 
certainly  taken  by  the  function  <j)(u)  as  u  is  made  to  increase.  Hence  the 
function  </>(u)  takes  the  value  £1  at  least  m  times  in  the  finite  portion  of  the 
plane  and  another  time  towards  infinity.  Since  by  hypothesis  <j>(u)  is 
indeterminate  for  u  =  oo,  it  appears  that  </>(u)  —  £o  is  zero  for  some  value 
of  u  such  that  u  <QO.  Call  this  value  am+i.  By  repeating  the  above 
process  it  may  be  shown  that  we  may  find  such  values  of  the  function  <f>(u) 
which  may  be  taken  arbitrarily  often  by  that  function. 

ART.  54.  We  may  derive  the  above  results  in  a  somewhat  more  explicit 
manner  by  means  of  our  eliminant  equation 


We  have  excluded  as  being  singular  all  values  of  the  function  £  =  <j>(u) 
which  satisfy  the  equation 

3/g,  P)  -  Q 

ar 

*  See  Weierstrass,  Zur  Theorie  der  eindeutigen  analytischen  Functionen,  Werke,  Bd.  II, 
pp.  77  et  seq.;  Weierstrass,  Zur  Functionenlehre,  pp.  1  et  seq.;  Hermite,  Sur  quelques 
points  de  la  theorie  des  fonctions,  Crelle,  Bd.  91,  and  "  Cours,"  loc.  cit.,  p.  98;  Mittag- 
Leffler,  Sur  la  representation  analytique,  etc.,  Acta  Math.,  Bd.  IV,  p.  8. 


PERIODIC    FUNCTIONS   IN   GENERAL.  65 

In  the  present  discussion  we  shall  also  exclude  the,  roots  of  the  equation 
/(£,  0)  =  0.  In  other  words,  the  function  £  =  (f>(u)  is  not  allowed  to  take 
those  values  of  u  which  make  £'  =  <f>'(u)  =  0. 

If  we  denote  by  fo  any  finite  value  that  (j)(u)  can  take,  then  all 
the  points  at  which  (f>(u)  can  take  this  value  £o  are  simple  roots  of  the 
equation  (f>(u)  —  £0  =  0;  for  this  difference  can  only  become  infinitesimally 
small  of  the  first  order  since 

<t>M  =  £o  +  ^  (u  -  u0)  +  ^  (u  -  u0)2  +  ----  , 

and  by  hypothesis  £0'  ^  0- 

It  follows  that  the  quantities  ab  a2,  .  .  .  ,  am  above  are  simple  roots 
of  the  equation  (f>(u)  —  £0  =  0,  and  consequently 


-f 

-».  J  c 


-fo 

if  the  integration  is  taken  over  a  closed  curve  in  the  (f>(u) -plane  that 
corresponds  to  a  circle  made  by  u  about  any  of  the  points  ai,  a2j  .  .  .  ,  am. 

We  also  saw  that  the  above  integral  indicates  the  number  of  circuits 
made  by  the  function  </>(u)  about  fo  m  the  ^(^)-plane.  As  this  integral 
equals  unity,  we  see  that  there  is  one  circuit  made  in  the  positive  direc 
tion  about  co  corresponding  to  the  circle  made  in  the  ?*-plane  about  any 
one  of  the  points  a.  All  values  c  i  which  belong  to  the  surface  included  by 
the  circuit  about  £o  are  therefore  taken  once  by  the  function  (j>(u)  if  u 
takes  all  values  within  the  corresponding  circle  Cft  about  au.  We  describe 
about  co  as  center  a  circle  C  with  so  small  a  radius  that  it  lies  totally 
within  the  above  circuit  S^  about  co-  We  shall  show  that  every  value  ci 
within  this  circle  is  taken  once  and  only  once  by  the  function  (j)(u)  when  u 
takes  all  possible  "values  within  the  circle  Cu. 

We  saw  that  the  integral 

d6(u) 


where  <j>(u)  is  regular  on  and  within  the  contour  C,  is  equal  to  the  number 
of  points  at  which  the  value  £  i  is  taken  within  C,  provided  each  point  is 
counted  as  often  as  the  order  of  the  zero  of  <j>(u)  —  £1  at  this  point.  It 
follows  under  the  given  hypotheses  that  the  above  integral  is  always  a 
positive  integer. 

If  then  £  is  considered  as  a  variable  complex  quantity  on  the  interior  of 
the  given  circle,  the  integral 


j_  r 

2*»J  < 


-  c 

is  an  analytic  function  of  £.  For  since  the  denominator  does  not  vanish 
for  any  point  on  the  periphery  of  the  circle,  the  elements  of  the  integral 
vary  in  a  continuous  manner  when  £  varies.  On  the  other  hand,  we  knovr 


66  THEOKY   OF   ELLIPTIC    FUNCTIONS. 

that  the  integral  is  equal  to  a  constant.  This  integral  considered  as  a 
function  of  £  must  also  be  equal  to  a  constant.  If  we  let  £  coincide  with 
£0,  the  integral  is  equal  to  unity.  It  follows  that  every  value  £i  which 
lies  sufficiently  near  £0  is  taken  once  and  only  once  if  u  remains  within 
the  circle  described  about  a^. 

We  draw  circles  as  indicated  above  around  all  the  points  ai,a2,  .  .  .  ,  am. 
These  points  are  the  values  of  u  which  cause  <J>(u)  to  be  equal  to  £0-  In 
the  </>(u) -plane  we  draw  the  corresponding  circuits  around  the  point  £o- 
Further  we  draw  a  circle  around  £0  as  a  center  with  a  radius  so  small  that 
it  lies  wholly  within  the  circuits  made  about  this  point.  Let  £1  be  a  point 
within  this  circle.  Then  the  value  £1  is  taken  by  <j)(u)  for  values  of  u 
once  in  each  of  the  circles  around  0,1,0,2,  .  .  .  ,  am  respectively  and  con 
sequently  at  least  m  times. 

We  consider  the  quantity 


where  </>(u)  takes  the  value  £0  at  the  points  u  =  a\,  a2,  .  .  .  ,  am. 

By  hypothesis  cf>(u)  —  £0  is  zero  of  the  first  order  on  each  of  these  points. 

By  Laurent's  Theorem  we  may  develop  -  in  the  neighborhood 


of  each  of  these  points;  and,  if  the  first  term  of  the  development  in  the 

neighborhood  of  alt  is  denoted  by  -  ^  —  ,  it  is  seen  that 

u  —a,, 


where  g(u)  has  the  character  of  an  integral  function  for  all  finite  values  of 
the  argument. 

Since  g(u)  cannot  be  a  constant,  as  otherwise  <j>(u)  would  be  a  rational 
and  not  a  transcendental  function,  it  is  seen  by  taking  values  of  u  sufficiently 
removed  from  the  origin  that  <j>(u)  —  £o  may  be  made  arbitrarily  small. 

Suppose  that  £1  is  a  value 'of  </>(u)  which  lies  within  the  interior  of  the 
circle  above.  It  is  clear  that  for  values  of  u  sufficiently  distant  from 
the  origin  the  function  <j>(u)  is  equal  to  £1.  We  have  also  shown  that 
besides  this  value  of  u  the  function  <f>(u)  takes  the  value  ci  at  m  other 
points  and  consequently  (/>(u)  takes  the  value  £i  at  m  +  1  points.  By 
continuing  this  process  it  may  be  shown  that  there  are  an  indefinite  number 
of  values  which  do  not  belong  to  the  singular  values  of  the  function  <p(u),  and 
which  may  be  taken  by  (j>(u)  an  arbitrarily  large  number  of  times.* 

It  follows  from  what  we  saw  in  Art.  38  that  <f>(u)  is  a  periodic  function. 

*  In  this  connection  see  Picard,  Memoire  sur  les  f auctions  enticres  (Ann.  EC.  Norm. 
(2),  9,  (1880),  pp.  145-166),  where  it  is  shown  that  an  integral  transcendental  function 
when  put  equal  to  any  arbitrary  constant  has  an  indefinite  number  of  roots  which 
are  isolated  points  on  the  w-plane. 


PERIODIC    FUNCTIONS    IN   GENERAL.  67 

ART.  55.     If  the  function  <j>(u)  has  the  properties  expressed  through 
the  equations 

/[*(*),  4^(10]  -  o, 

(v)^(u  +  v}\  =  0, 
v)  = 


we  have  seen  that  the  region  of  u  may  be  extended  by  analytic  continu 
ation  to  the  whole  plane  without  the  function  (f>(u)  ceasing  to  have  the 
character  of  an  integral  or  (fractional)  rational  function  for  all  values  of 
the  argument. 

If  <t>(u)  has  at  infinity  the  character  of  an  integral  or  (fractional)  rational 
function,  then  <j>(u)  is  a  rational  function  of  u;  but  if  the  point  at  infinity 
is  an  essential  singularity,  then  <j>(u)  is  a  periodic  function.  It  may  happen 
that  all  the  periods  may  be  expressed  as  positive  or  negative  integral 
multiples  of  the  same  quantity.  In  this  case  the  function  is  simply 
periodic  and  the  quantity  in  question  is  the  primitive  period  of  the  argu 
ment  of  the  function.  If  all  the  periods  of  a  function  can  be  expressed 
through  integral  multiples  of  several  quantities,  the  function  is  said  to 
be  multiply  periodic.  The  functions  with  two  primitive  periods  are 
called  doubly  periodic,  the  two  periods  constituting  a  primitive  pair  of 
periods. 

THE  PERIOD-STRIPS. 

ART.  56.  Consider  the  simple  case  of  the  exponential  function  eu. 
We  shall  first  show  that  eu  +  27:i  =  eu  for  all  values  of  u.  Writing  u  = 
x  -f  iy,  it  is  seen  that 

eu  _  ex  +  iy  _  g*(Cos  y  +  i  sin  y)  =  ex  cos  y  +  iex  sin  y. 

If  now  we  increase  u  by  2  -i,  then  y  is  in 
creased  by  2  x,  and  consequently  MO 


'277 


=  e*  cos  (y  +  Z  ~)  -h  ie^  sin  (y  - 
=  ex  cos  y  +  iex  sin  y  =  ex  +  iy  =  eu. 

It  follows  that  if  we  wish  to  examine  the 
function  eu,  then  clearly  we  need  not  study 

this  function  in  the  whole  ^-plane  but  onlv    ^^kx_L__ J 

o  x 

within  a  strip  which  lies  above  the  X-axis  Fig  4 

and  has  the  breadth  2  -.      For  we  see  at 

once  that  to  every  point  MO  which  lies  without  this  period-strip*  there 
corresponds  a  point  MI  within  the  strip  and  in  such  a  way  that  the  func 
tion  eu  has  the  same  value  at  MO  as  at  u\.  For  example  in  the  figure 


*  Cf.  Koenigsberger,  Elliptische  Functionen,  p.  210.  The  lines  including  a  period- 
strip  need  not  be  straight,  if  only  the  difference  between  corresponding  points  is  a 
period. 


68  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Suppose  that  p  =  a  +  ifi  is  an  arbitrary  complex  quantity,  and  con 
sider  the  equation 

eu  =  p  =  a  +  ifi. 

Let  us  first  see  whether  this  equation  can  always  be  solved  with  respect 
to  u;  and  in  case  it  is  always  possible  to  solve  it,  let  us  see  how  many 
values  of  u  there  are  within  the  period-strip  which  satisfy  it. 
We  have 

eU       =       6X     C0g      y       _|_       fox     gjn      y       =       p        _       a        _J_       ^ 

and  consequently 

ex  cos  y  =  a,  ex  sin  y  =  p. 

It  follows  that 


Since  a:  is  a  real  quantity,  the  positive  sign  is  to  be  taken  with  the  root. 
This  equation  determines  x  uniquely,  since  we  have  at  once 


x  =  log     a2  +  /?2. 

Q 

To  determine  y,  we  have  tan  y  =   £  • 

a 

Suppose  that  yo  is  a  value  of  y  situated  between  0  and  n  which  satisfies 
this  equation  (we  know  that  there  is  always  one  such  value  and  indeed 
only  one). 

It  follows  also  that 

tan  (y0  +  7i)  =  tan  y0. 

It  appears  then  as  if  y0  +  TT  satisfies  the  conditions  required  of  y.     This, 
however,  is  not  the  case,  since  we  have 

cos  (yo  +  TT)  =  —  cos  yo, 
sin  (y0  +  TI)  =  -  sin  ?/0, 

and  consequently  the  equations  ex  cos  y  =  a,  ex  sin  y  =  /?  are  not  satisfied 
by  the  value  y0  +  x. 

Hence  within  the  period-strip  the  equation 

eu  =  a  +  ifi 
is  satisfied  by  only  one  value  of  u  —  x  +  iy,  and  this  value  of  u  is 


u  =  log  V  a2  +  /?2  + 


On  the  outside  of  the  period-strip,  however,  the  equation  is  satisfied  by 
an  indefinite  number  of  values  of  u.  These  values  are  had  if  we  increase 
or  diminish  by  integral  multiples  of  2  m  that  value  of  u  which  satisfies 
the  equation  within  the  period-strip,  that  is,  if  we  keep  x  unchanged  and 
increase  or  diminish  the  value  ?/0  by  2  TT. 


PERIODIC    FUNCTIONS   IX    GENERAL. 


ART.  57.     We  shall  next  study  two  other  simple  functions,  cos  u  and 
sin  u.     These  functions  may  be  defined  through  the  equations 

cos  u  =  ±(eiu  +  e~iu), 

sinM  =  ^-.(eiu  -  e~iu}. 

£  i 
It  follows  at  once  that 

cos  (u  +  2  ~)  =  cos  u,         sin  (u  +  2  T:)  =  sin  u. 

Both  functions  have  the  period  2  TT.  We  may  therefore  limit  the  study 
of  these  functions  to  a  period-strip  with  breadth  2  x  measured  along  the 
lateral  axis. 

It  is  evident  that  to   every  point  MO  lying  without   this  period-strip 
there  is  a  corresponding  point  HI  within  the  strip  at  which  cos  u  and  sin  u 
have  the  same  values  as  at   MO.     For 
example  in  the  figure 

+  6  -)  =  COS  MI, 

+  6  -)  =  sin  MI. 


COS  UQ  =  COS 

sin  M    =  sin 


Suppose  next  that  p  is  an  arbitrary 
complex  quantity,  and  let  us  see  whether 
for  the  equation 

cos  u  =  p 


27T 


27T 


U0 


Fig. 


there  is  always  a  solution.  If  there  is  one,  there  is  an  indefinite  number. 
For  if  MI  satisfies  the  equation,  then  from  the  above  it  is  also  satisfied  by 
the  values  MI  +  2  r,  MI  -f  4  -,  •  •  •  . 

We  shall  show  that  there  are  always  two  values  of  u  within  the  period- 
strip  which  satisfy  the  equation. 

For  writing 

COSM  =  i(e*'"  +  e~iu}  =  p, 
we  have 

eiu  +  e-iu  =  2  p. 

Writing  eiu  =  t,  this  equation  becomes 

f*-2jX  +  l*-0.  (1) 

From  this  it  follows  that 


We  thus  have  two  values  of  t  =  e 
be  MI  and  u2,  so  that  therefore 

t\  =  cl'Ms     t2  =  el'"2. 
It  follows  that  we  have  for  iu\  and  iu2  values  of  the  form 


Let  the  corresponding  values  of  u 


u2  = 


where  k\  and  k2  are  positive  or  negative  integers. 


70  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

Dividing  by  i  we  have  at  once 

ui  =  —  iyi  +  ki2x, 
u2  =  —  if)2  +  k2  2  TT. 

Hence  clearly  there  are  two  solutions  of  the  equation  cos  u  =  p  within 
the  period-strip,  and  these  solutions  are  different  from  each  other. 
From  the  quadratic  equation  (1)  it  follows  that 

t1  .  t2  =  I,  oreiUleiU2  =  I. 
We  therefore  have 


and  consequently 

i(ui  +  u2)=  0  (mod.  2m), 
or 

HI  +  u2  =  0  (mod.  2  7t). 

i 

It  follows  that  the  two  values  of  u  which  satisfy  the  equation  cos  u  =  p 
within  the  period-strip  are  such  that  their  sum  is  equal  to  2  n. 

We  may  derive  similar  results  for  the  function  sin  u.  It  is  thus  seen 
that  the  two  functions  cos  u  and  sin  u  take  any  arbitrary  value  within  the 
period-strip  twice,  while  the  function  eu  takes  such  a  value  only  once  within 
its  period-strip. 

ART.  58.     The  period  a  of  a  simply  periodic  function  f(u)  is  in  general 

/a  complex  quantity.     We  have 
/(«+  -a)  -/(!»), 


#u 


/  and  if  we  write  u  =  0,  it  follows 

/  that 


f(a)  -  /(O), 


fl  I  that   is,  the   function  f(u)  has 

|  at  the  origin  the  same  value  as 
it  has  at  the  point  a  in  the  u- 
"L  plane;  and  also  at  the  points' 
.  .  .  ,  3  a,  2  a,  a,  -  a,  -  2  a, 
...  it  has  the  same  value  as  at 
the  origin. 

We  draw  through  the  origin 

an  arbitrary  straight  line  OL, 
and  through  the  points  a,  2  a} 
3  a,  ...,  —  a,  —  2  a,  ...  we 

draw  lines  parallel  to  OL.     The  entire  ^-plane  is  thus  distributed  into  an 
indefinite  number  of  strips. 

That  strip  which  is  made  by  OL  and  the  straight  line  through  +  a  par 
allel  to  OL  we  call  the  initial  strip. 


PERIODIC    FUNCTIONS    IN   GENERAL.  71 

Let  u  be  a  point  in  any  strip.  There  is  always  a  point  u'  in  the  initial 
strip  at  which /(w)  has  the  same  value  as  at  u.  For  if  through  the  point 
u  we  draw  a  line  parallel  to  the  line  that  goes  through  the  points  0,  a, 
2  a,  .  .  .  ,  and  on  this  line  measure  off  distances  a  until  we  come  within 
the  initial  strip  and  call  u'  the  end-point  of  the  last  distance  measured  off, 
then  u  and  u'  differ  only  by  integral  multiples  of  a,  so  that  the  function 
f(u)  has  the  same  value  at  both  points.  In  the  above  figure,  for  example, 
u  =  u'  +  2  a,  so  that/(w)  =  f(u'  +  2  a)  =  /(>')•  Hence  every  value  that 
the  function  can  take  in  the  w-plane  is  had  also  in  each  single  strip.  We 
therefore  need  investigate  every  simply  periodic  function  only  within  a 
single  period-strip.  This  we  have  done  above  for  the  simple  cases  of  eu, 
sin  Uj  cos  u. 

ART.  59.  If  a  represents  any  complex  quantity,  we  saw  in  Art.  26 
that  a  simply  periodic  function  with  a  as  a  period  may  be  readily  formed. 

2  JTt 

u 

Such  a  function  was  e  a     . 
Consider  next  the  series 

fr=+x  2  in 

<«-\  k v 

f(~.\  _     v    r,  P      a 

J  \  U )     -          f  f      lrt  &  ) 

where  the  constants  Ck  may  always  be  so  chosen  that  the  series  is  conver 
gent.*  It  is  clear  that  the  function  just  written  has  the  period  o;  and, 
since  the  constants  Ck  may  be  determined  in  different  ways,  it  is  clear  that 
an  arbitrarily  large  number  of  such  functions  may  be  formed,  all  of  which 
have  the  period  a. 
Such  a  function  is 

fc^X  fc^pu 

^=fe? ^T  =  0(w),    say, 


where  the  dks  are  also  constants. 

All  such  functions  have  the  property  that  there  is  no  essential  singu 
larity  in  the  finite  part  of  the  plane  and  they  are  indeterminate  for  no 
finite  value  of  u. 

For  the  point  u  =  oo  the  exponential  function  is  indeterminate  (Art.  21), 
and  for  all  other  values  of  u  it  is  seen  that  the  function  <j)(u)  is  one-valued. 

ART.  60.  Suppose  that  f(u)  is  a  one-valued  simply  periodic  function 
with  period  a  =  2  to,  and  which  has  only  polar  singularities  in  the  finite 
portion  of  the  plane. 

*  Cf.  Briot  et  Bouquet,  Fonctions  Elliptiques,  p.  161. 


71  THEORY   OF   ELLIPTIC    FUXCTIOXS. 

If  w*  put  «, 


em  =  *  = 
his  seen  that 

:r 


-  —  • 

T  ~ 

in  the  Mane,  when  0  varies  from  0  to  2  r,  the  variable  I  describes 
cte  about  the  origin  with  radius  r,  while  in  the  M-plane  the  variable  n 
the  straight  fine  AA'.  where  A  =  -  u»  log  r 
—  M*  log  r  +  2  w.     Furl  her,  when  *  varies  from 
2arto4r7  «  varies  from  ^' to  ^  where  again  .I'-i"  = 
2«»,  etc. 

Ifext  if  we  gjve  to  i  the  value  «*,  it  is  seen  that 
when  I  dLHUJJiB  m  circle  about  the  origin  in  the  /-plane 
with  radius  «,  m  deauBigii  the  straight  line  Bff.  where 

Hg.7.  It  foftows  that  m  the  t^plane  the  rectangle  AA'BB' 

uiucqponds  to  the  ringhiduded  between  the  two  circles 
with  radii  r  and  *  in  the  {-plane,  and  corresponding  to  the  initial  period- 
strip  m  the  it-plane  is  the  entire  l-plane.  Further,  any  period-strip  is,  as 
we  may  say,  c^ormaUy  represented  on  the  Ijianr  There  being  an  in 
definite  number  of  these  strips,  it  is  evident  that  to  any  value  of  /  in  the 
f^iaae  UHMUMBwlii  an  infinite  T^™J^^  of  ^*lpi»  ^  thr  •  piinr  differing 
:;v  i^:c-rril  :j.il::rles  :  2  _. 

Suppose  that  the  rectangle  AfRI*  m  taken  so  as  not  to  include  any 
of  the  •J-e-B"aS—  of /(«).  Then  if  F(l)  =/(M),  it  is  seen  that  F(f)  is 
regular  aft  al  points  at  which  /(«)  k  regular  and  consequently  may  be 
expanded  by  Lament's  Theorem  nt  a  aeries  of  the  font 


for  afl  values  of  i  aitnated  ••Hi'ii  the  ring-formed 

to  the  rectangle  AA'BB'. 
It  also  follows  that  for  aU  points  vttut  this  rectangle, /(*)  may  be 
ntm  convergent  series  of  the  form 

/(•)=  J  ^^ 

T.  =  —  X 

=  T^a,cosm^*  +  6*sinm^«^ 


PERIODIC    FUNCTIONS    IN    GENERAL.  73 

Prof.  Osgood,  loc.  cit.,  pp.  406  et  seq.,  gives  more  explicitly  the  limits 
within  which  such  series  are  convergent.* 

ART.  61.  We  next  propose  to  study  all  those  simply  periodic  functions 
which  first  are  indeterminate  for  no  finite  value  of  u,  which  therefore  in  the 
finite  portion  of  the  plane  have  no  essential  singularity,  while  they  are  inde 
terminate  for  u  =  infinity;  which  secondly  are  one-valued;  and  which  thirdly 
within  a  period-strip  take  a  prescribed  value  a  finite  number  of  times. 

Suppose  that  <f>(u)  is  such  a  function.  The  function  $(u)  behaves 
within  the  period-strip  in  a  similar  manner  as  do  the  rational  functions 
in  the  whole  plane.  For  if  w  =  4>(w)  is  a  rational  function  of  u7  then 
<&(u)  is  one- valued  and  for  every  given  value  of  w  there  is  only  a  finite 
number  of  values  of  u.  In  Art.  63  it  is  shown  that  at  the  end-points 
of  the  period-strip  the  function  has  definite  values. 

It  is  easy  to  see  that  the  function  (p(u)  which  we  are  considering  must 
be  indeterminate  at  infinity  in  the  direction  of  the  line  through  0,  a,  2  a, 
.  .  .  (see  Fig.  6).  For  let  u0  be  a  point  within  the  initial  period-strip. 
Draw  through  u0  a  line  parallel  to  the  line  through  0,  a,  2  a,  -  •  •  .  On 
this  line,  starting  from  u0?  we  measure  off  distances  a  an  indefinitely  large 
number  of  times.  We  thus  come  finally  to  infinity  and  the  function 
takes  at  the  end  of  the  last  distance  that  has  been  laid  off  the  value  <J>(UQ). 
Next  if  we  start  with  another  point  u\  and  proceed  to  infinity  in  the  same 
way  as  before,  the  function  will  take  for  the  infinitely  distant  point  the 
value  <£(MI).  Hence  at  infinity  there  appear  all  possible  values  which 
the  function  <f>(v)  can  take,  and  the  function  is  thus  said  to  be  indeterminate 
at  infinity  (cf.  Art.  3). 

ART.  62.  Let  w  =  <f>(u)  be  a  simply  periodic  function  with  the  period  a 
which  satisfies  the  three  postulates  made  above.  Further,  write 


so  that  t  and  ir  have  the  same  period  a  and  may  consequently  both  be 
considered  within  the  same  period-strip  of  the  w-plane.  Next  suppose 
a  given  value  is  ascribed  to  t.  Within  this  period-strip  there  is  (Art.  56) 
one  definite  value  of  u  which  belongs  to  the  prescribed  value  of  t.  If  we 
write  this  value  of  u  in  the  function  <£(u),  then  w  =  <j>(u)  has  a  definite 
value.  It  is  thus  shown  that  to  every  value  of  t  there  belongs  a  definite 
value  of  IT.  If  next  we  consider  not  only  one  period-strip  but  the  whole 
w-plane,  then  there  belongs  to  the  given  value  of  t  an  infinite  number  of 
values  of  M,  namely  in  each  period-strip  one  value.  And  if  u  is  one  of 
these  values  then  all  the  other  values  have  the  form  u  4-  A-a,  where  k  is 
a  positive  or  negative  integer.  If  we  write  all  these  values  in  6(u),  then 
IT  =  <£(M)  takes  always  the  same  value,  since  o(u  4-  ak)=  <r>(u).  Hence 

*  See  also  Henri  Lebesgue,  Lemons  sur  les  scries  trigonomctriques. 


74  THEORY   OF   ELLIPTIC    FUNCTIONS. 

also  when  we  consider  the  whole  w-plane,  for  every  definite  value  of  t 
there  is  one  definite  value  of  w.  Thus  we  have  shown  that  w  is  a  one- 
valued  function  of  t.  For  a  definite  value  of  w  there  are  after  the  third 
of  the  above  postulates  only  a  finite  number  of  values  of  the  argument  u 
in  each  period-strip.  Let  those  values  of  u  belonging  to  the  strip  in  ques 
tion,  be  ui,  u2)  .  .  .  ,  um,  and  let  the  corresponding  values  of  t  be 

2  «•                           2 «  2  jri 

—  u{  u2  —  um 

ti  —  e        ,     t2=  e  a     ,     .  .  .  ,    tm  =  e  a 

There  are  no  other  values  of  t  which  belong  to  the  given  value  of  w,  for 
if  we  extend  our  consideration  to  the  whole  w-plane,  that  is,  if  with  the 
given  value  of  w  we  also  associate  those  values  of  u  which  differ  from 
Ui,u2,  .  .  .  ,  um  by  integral  powers  of  a,  we  still  have  for  t  always  one  of 
the  values  ti,  t2,  .  .  .  ,  tm. 

We  have  previously  seen  that  to  each  value  of  t  there  belongs  only  one 
value  of  w.  We  now  see  that  to  every  value  of  w  there  belong  m  values 
of  t  and  therefore  that  t  is  an  m-valued  function  of  w.  It  follows  that 
w  and  t  are  connected  by  an  algebraic  equation  which  is  of  the  first  degree 
in  w  and  the  rath  degree  in  t,  say, 

F(w,  t)  =  0. 

Solving  this  equation  we  have 

w  =  <f  (0, 

•. 

where  ifr  denotes  an  algebraic  function  of  t. 

On  the  other  hand  we  saw  that  .w  was  a  one-valued  function  of  t,  and 
since  one-valued  algebraic  functions  are  the  rational  functions,  it  follows 

2m 
u 

that  w  is  a  rational  function  of  t  =  e  a    . 

We  have  then  the  important  theorem: 

Every  simply  periodic  function  <j>(u)  which  is  indeterminate  for  no  value 
of  u,  and  has  an  essential  singularity  *  only  at  infinity,  which  is  one-valued 
and  within  a  period-strip  can  take  an  ascribed  value  only  a  finite  number  of 

— M 
times  is  a  rational  function  of  t  =  e  a    ,  where  a  is  the  period  of  <j>(u). 

All  such  functions  may  therefore  be  written  in  the  form 

k  =  m  2xi 

-«— v  k  —  u 


W  = 


fc— « 

,     a 


fc=0 

where  the  Ck  and  dk  are  constants. 

*  A  treatment  of  simply  periodic  functions  which  have  essential  singularities  else 
where  than  at  infinity  is  given  by  Guichard,  Theorie  des  points  singuliers  essentiels 
[These,  Gauthier-Villars,  Paris.  1883J. 


PERIODIC    FUNCTIONS    IX   GENERAL.  75 

There  are  no  other  simply  periodic  functions  which  have  the  required 
properties. 

ART.  63.  We  may  make  m  and  n  equal  in  the  above  expression  without 
affecting  its  generality.  For  suppose  n  <  m.  We  have  then  to  put  all 
the  d's  in  the  denominator  equal  to  zero  from  dn+i  to  dm.  If  n  >  m, 
we  make  the  corresponding  •  change  in  the  numerator.  It  follows  that 
all  simply  periodic  functions  belonging  to  the  category  defined  above 
may  be  expressed  in  the  form 


k=0  k  =  0 

where  ^  is  a  rational  function  of  t.     Hence  the  points  t  =  ±  oo  ,  t  =  0 
are  not  essential  singularities  of  y]r(t)  and  consequently  also  </)(u)  has  definite 
values  for  u  =  ±  °o  .     In  other  words,  the  end-points  of  the  period-strips 
of  the  function  (f)(u)  are  not  essential  singularities. 
We  may  write  the  above  equation  in  the  form 

(cm  -  dmw)tm  +  (cm_i  -  dm_!  w)tm~l  +  '.  •  •  +  (c0-  d0w)  =  0, 

2m 

where  m  represents  the  number  of  values  which  t  =  e  a  can  take  for  a 
given  value  of  w,  or,  in  other  words,  the  number  of  points  in  each  period- 
strip  at  which  w  =  <j>(u)  takes  a  definitely  prescribed  value.  We  call 
m  the  degree  or  order  of  the  simply  periodic  function  w  =  (f>(u)  (cf.  again 
Art.  10). 

The  functions  cos  u  and  sin  u  must  be  expressible  in  the  above  form,  since 


u 


for  them  a  =  2  TT,  and  t  =  e  2*  =  eiu.  Further,  these  functions  take  a 
prescribed  value  twice  within  a  period-strip  (cf.  Art.  57)  and  are  conse 
quently  simply  periodic  functions  of  the  second  degree.  For  them  we 
must  have  m  =  2,  which,  indeed,  is  seen  from  the  relations 

1  /.   ,    l\      t2'+  Q.<  +  1 
cosu  =  }(*«•  -f  e-*)  -5V  +  l)=  2(0.<*  +  0   5 

rinu-  i  *>-  *-*  =  1  t* 


2i      0-t2  +  t 

Owing  to  the  relation  <j>(u)  =  ^(0  many  of  the  properties  of  simply 
periodic  functions  may  be  changed  into  properties  of  rational  functions; 
for  example,  the  function  <f)(u)  has  as  many  zeros  as  it  has  infinities  in 
each  period-strip.* 

*  Cf.  Briot  et  Bouquet,  Fonctions  Elliptiques,  p.  161;  Forsyth,  loc.  cit.,  p.  215; 
Osgood,  loc.  cit.,  p.  409;  Burkhardt,  Analyt.  Funktionen  einer  komplexen  Verdnder- 
lichen,  p.  161. 


76  THEORY   OF   ELLIPTIC   FUNCTIONS. 

THE  ELIMINANT  EQUATION. 
ART.  64.     In  the  case  of  the  function  eu  it  is  seen  that  if 

w  =  eu,  then  ~r  -  u  =  0; 
du 

and  if  w  =  cos  u,  then  {t£f  -  (1  -  w2)  =  0, 

\dul 

the  latter  differential  equation  being  satisfied  also  if  w  =  sin  u.  We  note 
that  these  three  functions  have  the  characteristic  that  each  of  them 
satisfies  a  differential  equation  in  which  the  independent  variable  u  does 
not  explicitly  appear. 

From  the  previous  article  we  saw  that,  if  w  is  a  simply  periodic  function, 
then 

w  =  </>( 

where  ^  is  a  rational  function. 

2m, 

Further,  since  e  a     =  t,  we  have 

^=r(t)  , 

du  .  a 

where  ^i  is  also  a  rational  function. 

By  eliminating  t  from  the  two  expressions  we  have  the  eliminant  equa 
tion  (Art.  34) 


where  /  denotes  an  integral  algebraic  function. 

In  Art.  41  we  said  that  if  there  existed  an  eliminant  equation  for  a 
one-valued   function   w  =  </>(u),   then   <f>(u)    had   an   algebraic   addition- 
theorem  and  belonged  to  one  of  the  categories  of  functions 
I.    Rational  function  of  u,  or  v 

2rt 

-  11 

II.    Rational  function  of  e  a      (simply  periodic),  or 
III.    Doubly  periodic  function. 

In  his  Cows  d'  Analyse  a  VEcole  Poll/technique,  in  1873,  Hermite  observed 
that  if  the  equation 

dw 


admits  a  one-valued  integral  (that  is,  if  w  is  a  one-  valued  function  of  u), 

we  may  express  w  and  —  rationally  in  terms  of  an  auxiliary  variable  t, 
du 

if  the  integral  w  is  a  rational  function  of  u,  or  if  it  is  a  simply  periodic 

function  of   u;  and  that  w  and   —  may  be  expressed  through  formulas 

du 


PERIODIC    FUNCTIONS   IN   GENERAL.  77 

which  include  no  other  irrationalities  than  the  square  root  of  a  polynomial 
of  the  fourth  degree,  if  w  is  a  doubly  periodic  function.* 

ART.  65.     The  following  question  arises:  What  further  conditions  must 

be  satisfied  in  order  that  an  integral  of  the  equation  f  I  w,  —  }  =  0,  belong  to 

\     du/ 
the  category  of  functions  defined  in  Art.  61? 

Such  a  function  must,  as  we  have  already  seen,  be  expressible  as  a  rational 
function  of  t,  say  "^(t),  and  its  derivative  is  also  a  rational  function  ^i(t). 

If  we  put  —  =  v,  the  above  equation  is 
du 

f(w,  v)  =  0. 

We  may  regard  this  integral  algebraic  equation  as  the  equation  of  a 
curve.  Strictly  speaking,  however,  this  can  only  be  done  if  w  and  v 
are  real  quantities;  still  we  may  speak  of  a  curve,  for  the  sake  of  a  graphical 
representation,  even  if  as  here  w  and  v  are  complex  quantities.  From 
what  was  shown  above,  if  we  write  for  w  a  certain  rational  function  ^(0 
and  for  v  a  rational  function  ^i(£),  the  equation  f(w,  v)  =  0  must  be 
identically  satisfied  for  all  values  of  t.  We  may  therefore  express  w  and 
v  rationally  through  a  parameter  t  in  the  form  of  the  equations  w  = 
^(0,  v  =  tyi(t).  Curves  in  which  such  a  rational  representation  of  the 
variable  t  is  possible  are  known  as  unicursal.^ 

If  then  an  integral  of  the  differential  equation 


is  to  belong  to  the  category  of  functions  which  we  are  studying,  the  equa 
tion 

f(w,  v)  =  0 

must  represent  a  unicursal  curve. 

But  this  condition  is  not  sufficient.  For  if  f(w,  v)  =  0  represents  a 
unicursal  curve,  there  is  an  infinite  number  of  ways  in  which  w  and  v  may 
be  expressed  rationally  in  terms  of  t.  But  among  these  ways  there  is 
one  which  is  such  that  t  for  every  prescribed  pair  of  values  of  w  and  v 
takes  only  one  definite  value.  Further,  if  w  is  a  function  of  our  category, 

it  must  be  a  one-valued  function  of  u,  and  consequently  0  *»  —  is  also  a 

du 

one-valued  function  of  u. 

But  if  w  and  v  are  given,  there  is  (as  we  have  just  seen)  only  one  value 
of  t  which  can  be  associated  with  them.  Hence  if  w  is  a  function  of  our 

*  Cf.  Cayley,  Lond.  Math.  Soc.,  Vol.  IV  (1873),  pp.  343-345. 

t  The  name  is  due  to  Cayley,  Comptes  rendus,  t.  62,  who  derived  the  funda 
mental  properties  of  these  curves.  See  also  Clebsch.  Ueber  diejenigen  ebenen  Curven 
deren  Coordinaten  rationale  Funktionen  eines  Parameters  sind,  Crelle,  Bd.  64. 


78  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

category,  the  parameter  t  must  be  a  one-valued  function  of  u.     Further, 
since 


it  follows  that 

*.  _  ±iOO  _  R(t) 
du      *'(0         ()' 

t 

where  R  is  a  rational  function  of  t. 

We  have   consequently  established  the  following:  The  integrals  of  the 
differential  equation 


may  be  functions  of  our  category,  first,  if  the  equation  f(w,  v)  —  0  represents 
a  unicursal  curve;*  second,  if  w  and  v  are  such  rational  functions  of  a 
parameter  t  that  to  every  pair  of  values  ofw,  v  there  belongs  only  one  value  of  t', 

and  third,  if  the  parameter  t,  as  determined  through  the  equation  f  Iw,  —  )  =  0, 

V      du/ 
is  a  one-valued  function  of  u.     It  does  not,  then,  necessarily  follow  that 

these  integrals  are  simply  periodic,  for  they  may  be  rational  functions  of  u. 
ART.  66.     The  parameter  t  determined  from  the  differential  equation 

£  =  R(t) 
du 

must  be  a  one-valued  function  of  u. 

We  are  thus  led  to  the  question:  What  is  the  nature  of  the  function  R(t), 
that  t  be  a  one-valued  function  of  uf 

If  we  consider  first  the  differential  equation 

where  the  g's  are  integral  functions  of  t,  the  condition  that  an  integral  t 
of  this  equation  be  a  one-  valued  function  of  u  is  that  g0  be  of  the  0  degree, 
0i  of  the  2d  degree,  g2  of  the  4th,  .  .  .  ,  gm  of  the  2  rath  degree,  f 

We  shall  derive  these  results  for  the  case  m  =  2  in  Chapter  V,  and  from 
this  it  will  be  seen  in  the  simple  case  before  us,  viz., 


du 

*  A  simple  method  of  representing  w  and  v  as  rational  functions  of  a  parameter  t, 
when  this  ca"n  be  done,  is  given  by  Nother,  Math.  Ann.,  Bd.  Ill;  see  also  Liiroth, 
Math.  Ann.,  Bd.  IV. 

t  Cf.  Forsyth,  loc.  cit.,  p.  481,  where  other  references  are  given. 


PERIODIC    FUNCTIONS   IN   GENERAL.  79 

that  t  is  a  one-valued  function  of  u,  if  R(t)  is  a  rational  integral  function 
of  the  2d  degree  in  t.  It  then  we  write  R(t)  =  a0  +  a  it  +  a2t2,  it 
follows  that 

r        dt 

J  aQ  +  ai$  +  «2^2 
where  7-  is  a  constant. 

We  have  four  cases  to  consider: 

(1)  Suppose  that  a2  7^  0  and  that  the  roots  of  the  equation  O,Q  +  atf 
•f  a2t2  =  0  are  not  equal. 

We  may  then  write  the  above  integral  in  the  form 


u+r=r * ._J rp u 

J  a2(t  -  a)  (t  -  P)       a2(a  -  P)  J   [t  -  a.       t  -  p J 

=  ^~?ilog 

It  follows  that 


a2(a  -/3)         t  -  a       t  -  p     * 
t-a 


_   -02(a-0)  (u  +  y) 

- 


and  consequently  t  may  be  determined  rationally  in  terms  of  an  exponential 
function  of  u.  Since  w  =  ^(0,  where  ^  is  a  rational  function,  it  is  seen 
that  in  this  case  w  is  a  rational  function  of  an  exponential  function  and 
therefore  belongs  to  our  category  of  functions. 

(2)  Suppose  that  a2  ^  0  and  that  the  roots  of  the  equation  a0  4-  <M 
+  a2t2  =  0  are  equal. 

We  then  have 


=  r 

J 


a2(t  -  a)2       a2(t  -  a) 

It  is  seen  that  in  this  case  /  is  a  rational  function  of  u,  and  since  w  is  a 
rational  function  of  t,  w;  is  a  rational  function  of  u  and  does  not  belong  to 
our  category  of  simply-periodic  functions. 
(3)  Suppose  that  a2  =  0.     We  then  have 


f 

J 


-log  (a0 


It  follows  that  a0  +  <M  =  eai(~u+y\  so  that  w  belongs  to  our  category  of 
functions. 

(4)  Suppose  that  a2  =  0  =  a^     It  is  evident  then  that 

Cdt        t 

u  +  T  =  I  —  =  —  > 
J   «o      ao 

or  t  =  a0(u  +  7-). 

In  this  case  w  is  not  a  simply-periodic  function. 


80  THEORY   OF   ELLIPTIC    FUNCTIONS. 

EXAMPLES 
1.   Consider  the  differential  equation 


..  dw 

or,  if  /.  v  =  —  -> 

du 

f(w,v)  =  w*  -(w-  v)3  =  0. 

We  must  first  determine  whether  this  equation  represents  a  unicursal  curve. 

If  we  write  w  —  v  =  tw, 

then  is  w*  -  w3t3  =  0, 

or  w  =  t3  -  ^(0  ; 

and  v  =  w(l  -  t)  =  ?(l  -  0  =  <Ai(0- 

It  is  thus  seen  that  w  and  t;  may  be  rationally  expressed  through  t. 

We  must  next  see  whether  t,  as  thus  determined,  has  a  definite  value  when  w 
and  v  have  prescribed  values. 

Since  w  =  t3,  to  one  value  of  w  there  correspond  three  values  of  t,  but  only  one 
of  these  can  satisfy  the  relation  v  =  w(l  -  £)>  when  a  fixed  value  is  given  to  v. 
Hence  to  every  pair  of  values  w,  v  there  belongs  a  single  definite  value  of  t.  We 
further  have  ji'(t)  =  3  t2  and 


du      V(t)  3? 

It  follows  that 


This  is  the  first  case  considered  above  where  a2  =  —  ^,  «  =  0,  /?  =  1. 
Integrating  we  have 


or 


I  _    e$(u+y)> 

all(*  W    =    t3     = ei(u+      )-.3  ' 

It  is  thus  shown  that  w  belongs  to  the  category  of  functions  considered. 
2.    Determine  the  integrals  of  the  differential  equation 


'duj      \du        I        \du 

that  is,  of  dw 

f  (w,  v}=  (v-  2)2  +  (v  -  l}w2  =  0,       where  v  =  — 

du 

It  follows  that  9L=2W(V-  1)  =0, 


SI  =  2(v-  2)  +  w2 
dv 

and  consequently  w  =  0,  v  =  2  is  the  double  point. 


PERIODIC   FUNCTIONS    IN    GENERAL.  81 

Hence,  if  we  write  w  =  (v  —  2)t,  it  follows  from  the  equation  of  the  curve  that 

1  +  (v  -  l)t2  =  0, 

or  v  =  1  -  \  =  ^(0 

and  w  =  -  t =  i,KO- 

The  curve  is  therefore  unicursal;  and  further  to  every  value  of  v  there  belong 
two  values  of  t,  but  of  these  only  one  can  satisfy  the  equation  for  w  when  to  v  a 
fixed  value  is  given. 

It  is  also  seen  that 

B0.fa.ML.-!, 

du      ^'(0 

and  consequently  we  have  the  fourth  case. 
It  follows  that  t  =  -  (u  +  f) 

and  w  = \-  u  +  f, 

u+  r 

and  being  a  rational  function  of  u  does  not  belong  to  our  category  of  functions. 
3.   Show  that  the  integrals  of  the  differential  equation 


J  \w'  T~         ~ 
\     du 

are  simply  periodic  functions. 
Note  that  the  equation 

f(w,  v)  =  (v-  2)2  (t;  +  1)  =  (3  w2  +  2)2 

is  satisfied  by 

v  =3(1  +  t2)  (1  +  3t2), 

w  =  3(t  +  t3). 
4.    Show  that  the  integrals  of 


\du 


3_  /A£V 

)        \du) 


are  rational  functions  of  u.  [Briot  et  Bouquet.] 

5.    Show  that  the  integrals  of 


du)  ~ 
are  simply  periodic  functions  of  u.  [Briot  et  Bouquet.] 


CHAPTER   IV 


DOUBLY  PERIODIC  FUNCTIONS.     THEIR  EXISTENCE. 
THE  PERIODS 

ARTICLE  67.     Returning  to  the  exponential  function  e*u,  we  know  that 
^  =  2  w,  say,  is  its  period. 


The  constant  /i  is  taken  real  or  complex  and  different  from  zero  or 
infinity.  tm 

Write  t  =  e^  =  e  w  ,  and  consider  the  function  <£(w)  =  ^(t),  where  here 
ty  is  not  necessarily  a  rational  function. 

Draw  the  period-strip  as  in  the  figure  and  let  u  be  any  point  within  or 
on  the  boundaries  of  this  strip. 

Let  \u    be  r  and  |  2  a>  \  be  s,  so  that 


2  CD       se10      s 

=  -[cos  (if/  -  0)  +  i  sin  (if/  -  0)]. 

If  R  denotes  the  real  part  of  the  complex 
quantity  after  it,  then  is 

--WcosOA-0)=^. 
2  ajJ      s  s 


Fig.  8. 


Hence  for  all  values  u  within  the  period-strip  we  have 


t*>. 

We  assume  that  <f)(u)  =  <f>(u  +  2  to)  and  that  <f>(u)  has  the  character  of 
an  integral  or  (fractional)  rational  function  for  all  points  within  the  period- 
strip  except  the  two  points  ±  oo. 

We  shall  show  (cf.  Art.  62)  that  if  <j>(u)  is  a  one-valued  function  of  u,  it 
is  also  a  one-valued  function  of  L  Let  u\  be  a  point  within  the  period- 
strip.  We  therefore  have  in  the  neighborhood  of  ui 


-  G 


P(u  - 


(A) 


where  G  denotes  an  integral  function  of  finite  degree  (including  the  Oth 
degree)  and  where  P  is  a  power  series  with  positive  integral  exponents. 

82 


DOUBLY   PERIODIC    FUNCTIONS.  83 

Ui>n  ,  .  iri 

(u-ti,)  -          / 

Let  ti  =e  ",  so  that  e          "=-; 

t\ 

f  (M-tti)  - 

further,  write  —  =  1  +  r  or  e  w  =  1  +  r. 

*i 
It  follows  that 

(u  -  ui)  ^  =  log  (1  +  r)  =  r  -  ir2  +  IT*  -  .   .  •  - 
oj  23 

This  series  is  convergent  for  values  of  r,  such  that 

0<  |T|<  1. 

But  we  had  r  =  -  -  1  =  ^-^  . 

«i  <i 

If  then  \t  —  t\\  <  |  £1  |,  we  have  the  convergent  series 


+ 

3 

This  expression  for  w  —  ui  substituted  in  the  equation  (A)  shows  that 
the  function  <j>(u)  considered  as  a  function  of  t  is  one-  valued  and  has  the 
same  character  for  t  =  ti  as  it  has  for  u  =  MI. 

ART.  68.     With  regard  to  the  f  unction  ^(M)  =  TJr(t)  two  cases  may  arise  : 

(1)  the  two  points  t  =  0,  t  =  oo  may  be  regular  points  of  the  function. 
In  this  case  ^(t)  is  a  rational  function,  as  there  is  no  essential  singularity. 

(2)  At  least  one  of  the  points  t  =  0,  t  =  oo  may  be  an  essential  singularity. 
In  this  case  we  shall  show  that  the  function  (f>(u)  has  another  period  2  a>'  ',  say, 

and  we  shall  prove  that  the  ratio  —  —  is  not  a  real  quantity. 

2  a) 

We  must  show  that  within  the  period-strip  there  are  values  which  may 
be  taken  an  arbitrarily  large  number  of  times  by  <j>(u).  It  follows  then  as 
in  Art.  38  that  there  exists  another  period  2  &/. 

Let  £0  be  a  value  which  </>(u)  may  take.  This  point  may  lie  anywhere 
in  the  finite  portion  of  the  period-strip  excepting  the  singular  values  of  u 
defined  in  Art.  37. 

Two  cases  are  here  possible:  (1)  The  function  <f>(u)  =  ^(t)  may  take  the 
value  £0  an  arbitrarily  large  number  of  times.  The  theorem  is  then 
proved.  (2)  The  function  <£(w)  may  take  the  value  ?0  a  finite  number  of 
times,  say  m,  within  the  period-strip.  Let  the  corresponding  values  of 
t  be  ti,  t2,  .  .  .  ,  tm. 

In  the  neighborhood  of  any  one  of  these  points  develop  -  -  by 

r  (0  ~~  £o 
Laurent's  Theorem. 

Then  as  in  Art.  53  it  is  seen  that  the  absolute  value  of  this  expression 
surpasses  every  limit  for  values  of  t  as  we  approach  one  or  the  other  (or 


84 


THEOEY   OF   ELLIPTIC   FUNCTIONS. 


possibly  both)  of  the  points  t  =  0  or  t  =  oo.  There  are  then  values  £1, 
say,  in  the  neighborhood  of  £0  which  are  taken  by  (f>(u)  =  ty(t)  at  least 
m  +  1  times.  By  continuing  this  process  it  is  shown  as  in  Art.  38  that 
(/>(u)  must  have  another  period  2  aj'  and  consequently 

<l>(u  +  2  to)  =  (j>(u), 
(f)(u  +  2a)')=  <j)(u). 

ART.  69.     It  follows  at  once  from  the  development  of  <j>(u)  in  the  neigh 
borhood  of  u\  in  the  form  (Art.  53) 


that  there  are  no  points  in  the  immediate  vicinity  of  u\  at  which  (f>(u) 
has  the  same  value  *  (Art.  8)  as  it  has  at  u\.  We  may  therefore  draw  with 
HI  as  center  a  circle  with  radius  p  which  is  so  small  (but  of  finite  length) 
that  within  the  circle  the  function  (f>(u)  does  not  take  the  same  value  twice. 
Further,  since  <j>(u  +  2  w)  =  <f>(u),  it  is  evident  that  \2a)\  >  p,  where  p 
is  a  finite  quantity. 

The  point  in  the  w-plane  which  represents 
2  oj  we  call  a  period-point.  Since  2  a>'  is  also 
a  period-point,  it  is  evident  that 


2(0' 


and  as  above 


-  2o/)= 


2aj-2o)'     >  p 


Fig.  9. 


It  is  thus  shown  that  the  distance  between 
two  period-points  is  always  a  finite  quantity. 

It  is  also  evident  that  if  we  bound  any  arbitrary  but  finite  portion  of 
surface  (S)  in  the  w-plane,  there  are  only  a  finite  number  of  period-points 
within  this  surface. 

If  A  is  a  period-point  and  if  B  and  D  are 
the  next  period-points  to  A,  then  C,  the 
other  vertex  of  the  parallelogram,  is  also  a 
period-point.  From  what  we  have  just 
seen  this  parallelogram  has  a  finite  area.  If 
then  there  were  an  infinite  number  of  period- 
points  within  (S),  there  would  be  within  this 
area  (S)  an  infinite  number  of  parallelo 
grams  with  finite  area,  which  is  impossible.  Fig.  10. 


*  Cf.  Burkhardt,  Analyt.  Funkt.,  p.  124;  Forsyth,  loc.  cit.,  p.  59;  Osgood,  Lehr- 
buch  der  Funktionentheorie,  p.  398. 


DOUBLY   PERIODIC   FUNCTIONS. 


85 


ART.  70.  We  consider  the  following  question:  If  2  co  =  a  and  2  a/  =  b 
are  periods  of  the  function  F(u)  and  in  the  sense  that  they  are  not  inte 
gral  multiples  of  one  and  the  same  primitive  period,  is  it  possible  for  the 
point  b  to  lie  on  the  line  joining  the  origin  and  the  point  a? 

The  quantities  a  and  b  may  be  written  in 
the  form  a  =      ie 

and  consequently,  if  b  lies  upon  the  straight  line 
Oa,  then         ,  =  «     .  A  =  0  _i_  - 

j(  JT 

We  therefore  have 

u — 

?  =  ±  -'  Fig-  U. 

6  s 

that  is,  the  ratio  ^  is  a  real  quantity.    The  above  question  may  consequently 
0 

be  expressed  as  follows :  Can  the  quotient  of  two  periods  a  and  b  be  a  real 

quantity  f 

Suppose  this  were  the  case  and  that  the  point  b  lies  upon  the  line  Oa. 

The  quantity  a  is  either  a  primitive  period  or  it  is  not  a  primitive  period. 

If  it  is  not,  it  may  be  written  in  the  form  a  =  ma,  where  a  is  a  primitive 

period  and  m  an  integer.  We  also  know  that  |  a  \  >  p,  where  p  is  a  finite 
quantity.  We  measure  off  upon  the  line  Oa  in  the 
direction  of  the  point  b  distances  a  and  have  the 
points  a,  2  a,  .  .  .  ,  ka,  (k  +  l)a,  -  -  •  .  If  b  coin 
cided  with  one  of  these  points,  for  example  ka,  we 
would  have 

b  =  ka,     a  =  ma, 

which  is  contrary  to  our  hypothesis. 

It  follows  that  b  must  lie  between  two  of  the  dis 
tances  measured  off,  say  between  ka  and  (k  +  l)a. 

Since  both  b  and  ka  are  periods,  the  distance  b  —  ka 
is  also  a  period.  We  therefore  have 

I  6  -  ka  I  <  I  a  I  . 


Fig.  12. 


Writing  b  —  ka  =  a',  we  measure  off  this  new  period  along  the  line  Oa 
and  make  for  a  the  same  conclusions  as  we  did  above  for  b.  We  find  that 
a  —  la'  is  a  new  period,  where  I  is  an  integer.  This  period  is  such  that 

|  a  -  la'  \  <  a'. 

By  continuing  this  process  we  come  finally  to  periods  whose  absolute 
values  are  smaller  than  any  assignable  finite  quantity  p,  which  is  a  con 
tradiction  of  what  was  proved  in  Art.  69. 


86  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  have  thus  shown  the  following  :  //  the  quotient  -  is  real,  there  exists  a 

b 

primitive  period  of  which  a  and  b  are  integral  multiples.     If  a  and  b  are  two 
different  periods,  as  defined  at  the  beginning  of  this  article,  then  the  ratio 

—  cannot  be  real,  and  b  cannot  lie  upon  the  line  Oa. 
b 

ART.  71.     The  above  theorem  is  due  to  Jacobi  (Werke,  Bd.  II,  pp.  25,  26), 

who  proved  it  as  follows:    Suppose  first  that  the  ratio  -  is  rational  and 
,  a 

write  -  »  22,  where  p2  and  pi  are  integers  that  are  relatively  prime. 
a       pi 

It  follows  that 

b        a 

—  =  —  =  a,     say, 

P2         Pi 

and  consequently  6  =  p2a  and  a  =  pia.     To  show  that  a  is  a  period  we 
determine  two  integers  qi,  q2,  such  that 

+     22=  I- 


We  know  that  there  are  an  infinite  number  of  solutions  of  this  equation. 
Multiplying  by  a  we  have 

Piaqi  +  p2aq2  =  a, 
or  qid  +  q2b  =  a. 

Thus  a  is  composed  of  integral  multiples  of  the  periods  a  and  b  and  is 
consequently  a  period.  Consequently  (Art.  70)  a  and  b  cannot  be  con 
sidered  as  two  different  periods. 

Suppose  next  that  the  ratio  -  is  real  but  irrational.     In  the  theory  of 

a 

continued  fractions  we  know  that  if 

—  -,       n  +  l  are  consecutive  convergents,  then 

Un         Un  +  l 

L_  =  ]*   ,  where  e  <  1. 

2 


Un         Un  +  l         UnUn  +  l         Un 

Hence  if  we  expand  -  in  a  continued  fraction  and  if  ^~  is  the  nth  conver- 
gent,  then  is    '          a  °» 

;~?-F5.  or  *J>-r+-T- 

a         On         On  On 

Since  dn  may  be  made  indefinitely  large,  it  follows  that 

|  dnb  —  ~rna  |  <  p,  where  p  is  as  small  as  we  choose. 
Further,  since  dn  and  fn  are  integers,  the  left-hand  side  is  a  period.     This 

contradicts  what  was  given  in  Art.  69.     It  is  thus  seen  that  the  ratio  - 

a 

must  be  a  complex  quantity  *  (including  the  case  of  a  pure  imaginary). 

*  See  Pringsheim,  Math.  Ann.,  Bd.  27,  pp.  151-157;  Falk,  Acta  Math.,  Bd.  7, 
pp.  197-200;  W.  W.  Johnson,  Am.  Journ.,  Vol.  6,  pp.  246-253;  Fuchs,  Crelle,  Bd.  83, 
pp.  13  et  seq.;  Me>ay,  Ann.  de  I'Ecole  Norm.  Sup.  (3),  t.  1,  pp.  177-184. 


DOUBLY   PERIODIC   FUNCTIONS.  87 

ART.  72.     We  may,  however,  prove  that  if  the  ratio  of  any  two  periods 
is  real  it  is  also  rational.      For  let  2  a>2,  2o>i  be  any  two  periods  whose 

ratio  is  real.     The  ratio  —  —  may  always  be  taken  positive;  for  if  it  were 
2  MI 

negative  we  might  substitute  the  period  —  2  aj2  in  the  place  of  -f  2  a>2. 

We  lay  off  the  periods  2a>i,  4o>i,  6<t»i,  .  .  .  ;  2  a>2,  4  a>2,  6  a>2,  .  .  . 
upon  the  same  straight  line  (cf.  Art.  70). 

It  is  evident  that  2  «,2  =  2  «,«,,  +  2  a*, 

where  mi  is  a  positive  or  negative  integer,  and  2  cu3  <  2  0*1.     Similarly  we 
write  4o;2  =  2  m^i  +  2  o>4, 

w2  being  an  integer,  and  2  o>4  <  2  o»  j  . 
It  follows  that  2a,2  _  2  mifl>i 


6  o>2  —  2 

and  consequently  the  quantities  2  o>3,  2  o>4,  2  w5,  .  .  .  are  all  periods. 

There  are  two  cases  possible:  (1)  These  quantities  are  all  different;  or 
(2)  they  are  not  all  different.  Suppose  that  2  o>3,  2  w4,  .  .  .  are  all 
different,  and  consider  the  n  quantities  2  w3,  2  o>4,  .  .  .  ,  2  a>n  +  <2,  to  which 
we  also  add  2  wi,  in  all  n  +  1  quantities. 

Divide  the  distance  between  0  and  2&>i  into  n  equal  parts;  then,  since 
each  of  the  quantities  2  0^3,  2  w4,  .  .  .  ,  2  cjn  +  2  is  IGSS  than  2  ^i,  two  of  these 
quantities  must  lie  within  one  of  the  n  equal  intervals.  Let  these  two 
quantities  be  2  w&  and  2  a>i.  It  is  clear  that  2  Wfc  —  2  o>j  is  also  a  period 

and  less  than  —  —• 
n 

Since  n  is  an  arbitrarily  large  integer,  it  is  seen  that  we  have  here  periods 
that  are  arbitrarily  small,  contrary  to  what  was  proved  in  Art.  69.  It 
follows  then  that  two  of  the  above  quantities  must  be  equal  (which  includes 
now  also  the  second  case).  We  then  have  for  example 

2iOJq  +  2  ==  2  CJp  +  2) 

so  that  2  qa>2  —  2  mqa)i  =  2  paj2  —  2  mpa)i, 

mq  and  mp  being  integers;  and  from  this  it  is  seen  that  —  —  must  be  a 

rational  quantity. 

ART.  73.     We  mav  prove  as  follows  that  the  ratio  -  cannot  be  real. 

2aj 

For  take  in  the  period-strip  of  Art.  67  two  points  u2  and  u\  such  that 
u?  —  u\  =  2  a/.     In  that  article  we  saw  that 


and 


88 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


It  follows  that 

If  now 
then  is 


,/«2_I 
V        O 

\    * 


<  i. 


is  a  real  quantity, 


2w 
<  1,    or  2  a/  <  2a>. 


Fig.  13. 


We  thus  have  two  periods  which  lie  along  the  same  straight  line,  of  which 
one  is  less  than  the  primitive  period  2  a>,  which  contradicts  the  notion  of 
a  primitive  period.  Hence  2  w  and  2  a>'  must  have  different  directions.* 
ART.  74.  There  exist  two  primitive  periods  through  which  all  other 
periods  may  be  expressed. 

Geometrical  Proof. 

We  shall  first  show  that  it  is  always  possible  to  form  a  period-parallelo 
gram  which  is  free  from  periods.  Suppose  that  in  the  period-parallelo 
gram  formed  of  the  periods  a  and  b  there  are  present  periods.  Their 
number  must  be  finite  (Art.  69).  Among  all  these  periods  let  /?  be  the  one 

whose  perpendicular  distance  on  Oa  is  the 
shortest.  It  is  then  evident  that  the 
period-parallelogram  constructed  on  Oa 
and  O/?  is  free  from  periods.  Of  course 
we  have  assumed  that  Oa  is  not  an  inte 
gral  multiple  of  another  period. 

It  is  evident  that  7-  is  a  period  since 
a  +  /?  =  f]  and  it  is  also  evident  that  there  can  be  no  period-points 
within  or  on  the  boundaries  of  afiy. 

If  for  example  A  were  a  period-point  on  the  side  /??-,  then  through  A 
we  could  draw  the  parallel  to 
the  side  0/9  which  cuts  the  line 
Oa  in  jj..  We  would  then  have 
a  period-point  at  /i,  which  con 
tradicts  the  fact  that  no  period- 
point  lies  on  Oa. 

In  the  same  way  it  may  be 
shown  that  no  period-point  lies  Fig  14^ 

on  a?-. 

Suppose  next  that  a  period-point  v  lies  within  the  triangle  fifa  (Fig.  15); 
then  by  completing  the  parallelogram  ftvajj.  it  is  seen  that  JJL  is  also  a  period- 
point  and  lies  within  the  triangle  0/fo,  which  contradicts  what  we  saw  above. 

*  Picard,  Traite  d' Analyse,  t.  2,  p.  220,  gives  an  interesting  proof  of  this  theorem; 
see  also  other  proofs  in  Hermite's  "  Cours  "  (4me  e"d.),  p.  217,  and  Goursat,  Cours 
d' Analyse,  t.  2,  No.  314. 


DOUBLY    PERIODIC    FUNCTIONS.  89 

We  thus  see  that  within  the  entire  parallelogram  Opra,  the  sides  included, 
there  are  situated  no  period-points  except  at  the  vertices.  It  is  also  evident 
that  if  the  whole  u-plane  be  filled  with  the  congruent  parallelograms,  as 
indicated  in  Fig.  16,  there  is  nowhere  a  period-point  except  at  the  ver 
tices.  If  for  example  there  were  a  period-point  u  in  any  of  the  parallelo- 


o  a 

Fig.  15.  Fig.  16. 

grams,  there  exists  in  the  initial  parallelogram  Op  fa  a  point  uf  which  differs 
from  u  only  by  integral  multiples  of  a  period,  and  contrary  to  hypothesis 
there  would  be  a  period-point  within  the  initial  parallelogram.  It  is  also 
evident  that  the  vertices  of  all  the  parallelograms  are  period-points  since 
they  are  of  the  form 

ka  +  ip, 

where  k  and  I  are  integers. 

It  follows  that  a  one-valued  analytic  function  cannot  have  three  inde 
pendent  periods  a,  6,  c;  for,  as  we  have  just  seen,  these  three  quantities  are 
expressible  in  the  form 

a  =  ka  +  ip, 

b  =  k'a  +  I'p, 
c  -  k"a  +  l"p, 

where  the  fc's  and  I's  are  integers. 

We  have  thus  shown  that  a  one-valued  analytic  function,  which  (in  the 
neighborhood  of  at  least  one  point}  is  developable  in  an  ascending  integral 
power  series,  cannot  have  more  than  two  independent  periods. 

We  shall  see  later  that  the  pairs  of  primitive  periods  may  be  chosen  in 
an  infinite  number  of  different  ways  (see  Art.  80). 

ART.  75.  It  is  evident  from  the  foregoing  that  it  is  only  necessary  to 
consider  the  values  of  a  doubly  periodic  function  (f>(u)  within  the  initial 
period-parallelogram  whose  sides  are,  say,  a  =  2  a),  t3  =  2  a)'.  In  this 
parallelogram  the  function  <j>(u)  has  everywhere  the  nature  of  an  integral 
or  a  (fractional)  rational  function.  We  shall  agree  that  the  second  period 
lies  to  the  left  if  we  look  from  the  origin  toward  2  at.  (See  Fig.  17). 


90  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  may  write 

|^-  =    T    =    O    +   if), 

2  co 
where  by  hypothesis  \  p\  ^  0,  since  the  ratio  ~-  is  not  real.     All  points 

2oH-2w'  within  the  interior  and  on  the  sides  of  this 
period-parallelogram  may  be  expressed  in 
the  form 

u  =  2  tco  +  2  t'a)', 
o  2u>  where  0=2=1,     0  ±  Z'  ^  1. 

pig  17  The  totality  of  all  such  values  of  u  may 

be  considered  as  the  analytic  definition  of 

a  period-parallelogram.     The  vertices   (except   the   origin)  are  excluded 
from  the  consideration. 
Further,  let 

w  =  2  mco  +  2  m'aj' 

where  m  and  m'  are  real  quantities. 

It  follows  that  ~  * 

JtL=m  +  m>^. 
2  co  (o  ' 

and  since  —  is  a  complex  quantity,  -^-  is  also  complex,  =  of  +  ip'  ,  say. 
CD  2  co 

We  thus  have 

m'  (o  +  ^), 


or  <7r  =  m  +  m!  a,    pf  =  m'  p. 

It  follows  that 

m'=£-,      m  =  a'-  &-<*. 
p  p 

Since  p  is  different  from  zero,  the  denominator  does  not  vanish,  and 
consequently  m  and  m'  are  determinate  quantities. 

It  is  thus  seen  that  every  complex  quantity  w  may  be  uniquely  written 
in  the  form 

w  =  2  mto  +  2  m'co', 

where  m  and  m'  are  real  quantities. 

ART.  76.     Two  points  w  and  w'  are  called  congruent  if 

w  -  w'  =  2  kco  +  2  Ico', 

where  k  and  I  are  integers.     The  fact  that  w  is  congruent  to  w'  may  be 
written 

w  =  w'  (modd.  2  co,  2  to')  ; 

or,  if  no  confusion  can  arise, 

w  =  w'. 


DOUBLY   PEKIODIC    FUNCTIONS.  91 

It  is  also  clear  that,  when  w  and  w'  are  congruent,  then  w  —  w'  is  a  period 
of  the  argument  of  the  function. 
If  we  write 

w  =  2  rnu)  +  2  rn'oj', 
w'  =  2  nw   +2  n  V, 
and  if  w  =  wf  (modd.  2  co,  2  a/), 

it  is  evident  that  the  quantities  m  and  n,  as  also  the  quantities  m'  and  n', 
differ  only  by  integers,  that  is,  m  —  n  =  integer  as  is  also  m'  —  n'. 

ART.  77.  Suppose  that  the  period-parallelogram  formed  on  the  two 
sides  0  .  .  2  co  and  0  .  .  2  a>'  is  free  from  period-points.  We  may  show 
analytically  that  all  the  period-points  in  the  w-plane  are  composed  through 
addition  and  subtraction  of  2  a>  and  2  a/. 

For  let  2  aj     =  I. 

c\       / 

Then,  since  -  —  =  a  -f  ip, 

2w 

it  is  seen  that 

|  2  a/  | 

Further,  since  2  a>  +  2  a/  =  2a>(l  +  cr  +  1,0),  it  follows  that  the 
length  of  one  diagonal  of  the  parallelogram  is 

2  w  +  2  a/ 


while  the  length  of  the  other  diagonal  is 


Represent  by  L  the  longest  of  the  four  sides 

|  2  aj  |,  |  2  w'  |,  |  2  oj  +  2  w'  ,    |  2  a/  -  2  w  |. 

Next  divide  the  two  sides  0  .  .  2  a>  and  0  .  .  2  a/  respectively  into 
n  equal  parts,  so  that  the  period-parallelogram  will  be  divided  into  n2 
small  parallelograms.  The  distance  between  any  two  points  situated 

within  one  of  the  smaller  parallelograms  is  not  greater  than  —  • 

n 

If  there  are  periods  that  cannot  be  expressed  through  integral  multiples 
of  2  a;  and  2  a>'  and  if  2  a>i  is  such  a  period,  we  shall  construct  the  con 
gruent  point  which  lies  within  the  initial  period-parallelogram. 

We  ma    write 


where  0  =#1  <  1      and      0  =  ,«i'  <  1. 

This  point  must  fall  within  or  on  the  boundaries  of  one  of  the  small  parallel 
ograms. 

Admitting  (Art.  69)  that  every  period  has  a  definite  length,  it  may  be 
shown  as  follows  that  r«i  and  («i'  are  rational  numbers. 


92  THEORY    OF   ELLIPTIC   FUNCTIONS. 

We  have  the  congruence 

2  0)i   =  2  fJLiOJ    +  2  [JLl'o)'  ', 

and  in  a  similar  manner  we  form 

2  •  2  0)1  =  2  z2w  -f  2  *2'^' 


2(n2  +  l)o>!  = 

where  0  =  /^  <  1     and     0  =  /*/  <  1, 

(&  =  1,  2,  .  .  .   ,  n2  +  1). 

If  these  n2  +  1  points  in  the  initial  period-parallelogram  are  all  different, 
at  least  two  of  them  must  fall  within  or  on  the  boundaries  of  one  of  the 
small  parallelograms,  and  the  distance  between  these  points  is  therefore 

less  than  —  .      As  n  can  be  made  arbitrarily  large,  there  are  then  periods 
n 

that  are  arbitrarily  small,  which  is  contrary  to  our  hypothesis. 

It  follows  that  at  least  two  of  the  n2  +  1  points  must  coincide,  in  which 
event  we  would  have 

2  pcoi  =  2  /j.pO)  +  2  /ip'o)', 

2  qo)i  =  2  JUPOJ  +  2  /*P'a>', 
and  consequently    2(p  —  q)co\  =  0  (modd.  2  o),  2o>'), 

where  p  and  q  are  both  integers.  We  have  thus  shown  that  an  integral 
multiple  of  2  MI  is  congruent  to  the  origin.  Since  2  o),  2  a>'  are  a  pair  of 
primitive  periods,  it  follows  from  the  theorem  of  the  next  article  that 
2  a>  i  must  be  congruent  to  the  origin. 

ART.  78.     Jacobi    (Werke,   Bd.    II,    pp.    27-32)    proves    the   following 
theorem:   //  a  one-valued  function  has  three  periods  0)1,  0)2,  ^3,  such  that 
miaji  +  m2o)2  +  m3a)3  =  0, 

where  m\,  m^  niz  are  integers,  then  there  exist  two  periods  of  which  <DI,  o>2,  ^3 
are  integral  multiple  combinations. 

We  may  assume  that  there  is  no  common  divisor  other  than  unity  of 
mi,  ni2,  m^.  Let  d  be  the  common  divisor  of  m,2  and  m^.  Of  course, 
d  =  1  when  m2  and  w3  are  relatively  prime. 

Then,    since    —  a>i  =  —  7^aj2  —  —  6t>3   and  the  right-hand  side  is  an 

d  d  d  m 

integral  combination  of  periods,  it  follows  that  —  ^i  is  a  period.     Since 

CL 

—  is  a  fraction  in  its  lowest  terms,  when  expressed  as  a  continued  frac- 
d 

tion  it  may  be  written  mi  _  p  =  .    l_ 

d        q  dq' 

where  ^  is  the  last  convergent  before  the  proper  value.     It  follows  that 
3 

y~-aji  -  pcoi  =  ±  -wi  =  aj,  say, 
d  d 

where  oj  is  a  period. 


DOUBLY   PERIODIC    FUNCTIONS.  93 

Let  ^  =  m*>  ^='*3/> 

d  d 

so  that  mico  -I-  m2'a)2  +  m^'ais  =  0. 


Change^2-  into  a  continued  fraction,  taking-  to  be  the  last  convergent 

m3'  s 

before  the  proper  value,  so  that 


Z^2_  _  L  =  ± 


ra3        s  sm3 

Then  rco2  +  sa)3  being  an  integral  combination  of  periods,  is  a  period 
a*',  say. 

On  the  other  hand, 

±  co-2  =  a)2(sm2   —  7*7/13') 

=  —  nutria   —  s(mi<jj  4-  m^'aj^) 


also 


m2ajf', 


and  <^i  =  ^/tt». 

Hence  two  periods  &>,  o>r  exist  of  which  o>i,  o>2,  ^3  are  integral  multiple 
combinations.* 

We  may  conclude  from  the  foregoing  that  All  one-valued  analytic 
functions  are  either 

(1)  Not  periodic,  or 

(2)  Simply  periodic,  or 

(3)  Doubly  periodic. 

Triply  or  multiply  periodic  one-valued  functions  do  not  exist. 

ART.  79.  We  may  next  prove  the  following  theorem:  It  is  possible  in 
an  infinite  number  of  ways  to  form  pairs  of  primitive  periods  of  a  doubly 
periodic  function. 

Let  la).  2  a/  be  a  pair  of  primitive  periods,  and  suppose  that 

•     !|*.-.«  +  * 

2  CD 

where  p  is  positive,  that  is, 


We  wish  to  form  another  pair  of  primitive  periods  2  io,  2  o>'  such  that 


*  Cf.  Forsyth,  Theory  of  Functions,  p.  202;  see  also  Hermite  in  Lacroix's  Calculus, 
Vol.  II,  p.  370. 


94  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  is  evident  that  we  must  have 

2  aj  =  2  pco  +  2  quj', 
2at'  =  2p'a)  +  2q'a}', 

where  p,  q,  p',  qf  are  integers. 

Further,  p  and  q  must  be  relatively  prime,  for  otherwise  2  aj  would  be 
the  integral  multiple  of  a  period.  The  integers  p'  and  q'  must  also  be 
relatively  prime.  It  follows  that 


pq'  -  qpf 

Since  2  to  and  2  to'  are  to  be  a  pair  of  primitive  periods,  the  period  2 
must  be  expressible  integrally  through  them. 
It  follows  that 


and 


pqf  -  qp'  pqf  -  qp' 

must  be  integers. 
We  further  have 


pq'  -  qp' 


c  uentl 


and         f  P are  integers. 


pcf  -  qp'  pq'  -  qp 

If  we  put  pq'  —  qp'  =  A,  it  is  seen  that  the  four  quantities  above  are 
integers,  if  A  =  ±1.     For  suppose  that  A  is  different  from   ±1.     It 

would  then  follow,  since  ^-  and  -2-  are  to  be  integers,  that  q  and  p  have  a 

common  divisor  other  than  unity,  which  is  contrary  to  the  hypothesis. 
The  next  question  is:  Are  both  values  A  =  -fl  and  A  =  —  1  admissible? 
We  required  that 

0     and     R 


We  have 

257        2  y/qj  +  2  9  V 


h 


Since  —  =  a  +  ip,  it  follows  that 


2  5i       i[p  +  9(cr  +  ip)]  (p 

and  consequently 


ip)         ~(p'  +  q'o}qp  +  (p  .r      , 


%*/%)       (p  +  qa)2  +  q2p< 


DOUBLY   PERIODIC   FUNCTIONS.  95 

As  p  is  positive  by  hypothesis,  we  must  have  pq'  —  qpf  positive  in  order 
to  fulfill  the  condition 

It  follows  then  that 

A  -  pq'  -  qp'  -  +  1. 

ART.  80.  Using  the  condition  just  written,  we  may  form  an  arbitrary 
number  of  equivalent  pairs  of  primitive  periods  as  soon  as  one  such  pair 
is  known.* 

The  transition  from  one  pair  of  periods  to  another  is  known  as  a  trans 
formation,  and  the  quantity  A  =  pq'  —  qp'  is  called  the  degree  of  the 
transformation.  We  have  here  to  consider  transformations  of  the  first 
degree. 

The  quantity  A  gives  the  measure  of  the  surface-area  of  the  second 
period-parallelogram,  if  that  of  the  first  is  denoted  by  unity. 

Hence  all  primitive  period-parallelograms  have  the  same  area,  for  if 

2  5  =  x  +  iy     and     2  a/  =  x'  +  iy*, 
the  area  of  the  corresponding  parallelogram  is 

±  (xi/'  -  ?/.r')- 
If  further, 

2  cu  =  c  +  iy     and     2  u>'  =  £'  +  iif, 

the  area  of  the  corresponding  period-parallelogram  is 

It  follows  that,  if 

2  5  =  2  poj  +  2  quj'     and     2  at'  =  2  p'aj  +  2  q'a)', 

,  x  =  p;  +  q;'.  (  x*  =  p'*  +  gT, 

then  I  and     { 


y  =  py  +  gy'; 

and  consequently 


But  here  pq'  -  qp'  =  1. 

Hence  a  primitive  period-parallelogram  is  not  unique. 

The  linear  substitution 

2  Z>  =  2  paj  +  2  qa>', 

2  5'  =  2  p'a)  +  2  q'a)' 
is  denoted  by 

•P, 
>',  3'. 

*  Cf .  Briot  et  Bouquet,  Fonctions  Elliptiques,  pp.  234,  235,  and  pp.  268  et  seq. 


96 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


One  of  the  substitutions  which  satisfies  the  condition 

A  =  pcf  -  qp'  =  1 


r  *iii 

I-  i ,  oj 


18 


In  this  case  we  have 

2a)  =  2cof, 

2  £'  =  -2  co. 
A  second  substitution  which  satisfies  the  same  condition  is 


or 


2  a)  =  2  co, 
2  £'  =  2aj 


2  a/ 


It  may  be  shown  that  every  linear  substitution  with  integral  elements 
and  determinant  A  =  1  may  be  formed  by  a  finite  number  of  repetitions 
of  these  two  substitutions. 

ART.  81.  The  question  arises*  whether  among  the  infinite  number 
of  equivalent  pairs  of  periods  there  are  those  to  which  preference  should 
be  given.  There  are  one,  two,  and  sometimes  three  pairs  of  primitive 
periods  which  may  be  chosen  in  preference  to  the  others.  One  of  the 
periods  in  these  selected  pairs  of  periods  has  the  smallest  absolute  value 
among  all  the  periods.  It  is  clear  that  such  a  period  exists;  indeed  there 
are  two  such  periods  differing  only  in  sign.  Taking  this  smallest  period 
as  a  radius  we  describe  a  circle  about  the  origin.  Within  this  circle  no 
period  can  be  situated,  but  upon  the  periphery  there  lie  at  least  two 
periods  (180  degrees  from  each  other).  It  is  also  seen  that  the  surfaces 
of  the  two  circles  drawn  about  these  period-points  and  having  the  same 

radii  as  the  first  circle  must  be  free 
of  periods.  Hence  besides  the  period- 
points  P  and  Pf  none  can  be  situated 
on  any  part  of  the  periphery  of  the 
first  circle  except  the  shaded  arcs  P\P% 
and  PsP±.  On  these  arcs  there  may 
be  two  periods  differing  by  180  degrees 
and  possibly  four  periods. 

In  the  last  case  the  period-points 


Fig.  18. 


must  lie  at  the  four  points  of  intersection  of  the  circles,  viz.,  Pi}  P2,  PS 
and  P4,  so  that  there  may  lie  upon  the  first  circle  two,  four,  or  at  most 
six  period-points;  and  consequently  the  period  of  smallest  absolute  value 
is  either  2-ply,  4-ply,  or  6-ply  determined. 

*  Cf.  Burkhardt,  Elliptische  Funktionen,  p.  194. 


DOUBLY   PEKIODIC    FUNCTIONS. 


97 


Denote  any  one  of  these  six  periods  by  2  o>,  which  we  use  as  one  of  the 
selected  pair  of  primitive  periods. 

We  shall  impose  a  further  condition  upon  the  other  period  of  this  selected 
pair.  The  second  period  2 a/  must  lie  to  the  left  of  0  .  .  2a>.  We  also 
know  that  |  2  a/  |  >  \  2  a>  \ .  We  cut  a  strip  out  of  the  plane  as  indicated 
in  the  figure.  The  second  period-point  may 
always  be  made  to  lie  within  this  strip ;  for  if 
it  were  situated  without  the  strip,  by  the 
addition  of  2mw,  where  m  is  a  positive  or 
negative  integer,  it  can  be  caused  to  lie  within 
the  strip,  but  it  does  not  fall  within  that  part 
of  the  strip  which  belongs  to  the  two  circles. 
Hence  the  triangle  0  .  .  2  to  .  .  2  ujf  has  only 
acute  angles,  the  right  angle  being  a  limiting 
case. 


We  write      r  =  = 


2co 


Fig.  19. 


where 


<  1. 


Owing  to  the  substitution 


we  may  so  choose  2  to'  that 

It  follows  that  t3  >  i  V§. 
If  further  we  write 

h  =  q  =  eTri  = 
it  is  clear  that  I  <    _  v/3    < 

a  fact  which  we  shall  find  to  be  very  important  in  the  development  of  the 

Theta-functions  (Chapter  X). 

ART.  82.  We  have  interpreted  the  equa 
tion  A  =  pq'  —  qp'  =  1  as  denoting  that  the 
parallelograms  formed  on  pairs  of  primitive 
periods  have  the  same  area.  Let  2  5,  2  5' 
be  a  pair  of  primitive  periods.  The  quan 
tities  2  to  and  2  to'  determine  a  triangle,  and 
all  such  period-triangles  have  the  same  area. 
Let  1251=7. 


2Z2T 


Fig.  20. 


3l2 


Then    if 


^  =  «  +  #, 
9  (it 


the  area  of  the 


co 


triangle  is  ^-  and  that  of  the  period-parallelogram  is  /9/2.     This  quan- 
& 

tity  being  constant  for  all  equivalent  primitive  pairs  of  periods,  we  have 

const. 


0- 


I2 


98  THEORY   OF   ELLIPTIC   FUNCTIONS. 

From  this  it  is  seen  that  /?  is  a  maximum  when  I  is  a  minimum.  If  then 
P  is  to  have  its  greatest  value,  we  must  choose  the  first  period  2  a>  so  that 
it  has  the  smallest  possible  value. 

If  the  ratio  of  the  periods  is  a  pure  imaginary,  then  a  =  0  and  /?  =  1  . 
In  this  case 


EXAMPLE 

If  ton  co2  and  WA  are  periods  of  <j>(u)  and  if 

29w3  =  17  0)^  +  11  <*}2, 

show  that 

co  =  5  co  !  +  3  w2  —  8  w3 

w'  =  3  ft>!  +  2  w2  -  5  w3 
are  a  pair  of  primitive  periods  of  <j>(u).  [Forsyth.] 


CHAPTER  V 
CONSTRUCTION  OF  DOUBLY  PERIODIC  FUNCTIONS 

Hermite's  Intermediary  Functions.     The  Eliminant  Equation. 

ARTICLE  83.  Having  established  the  existence  of  the  doubly  periodic 
functions,  we  shall  next  show  how  to  construct  such  functions  and  natur 
ally  the  simplest  ones  possible. 

The  expression 


k  =  +  »  .  2  ?n 

V     i     *  — u 


is  a  simply  periodic  function  which  can  be  developed  in  positive  ascending 
powers  of  u  —  UQ,  and  which  is  not  indeterminate  or  infinite  for  any  finite 
value  of  u,  provided  the  constants  At  have  been  suitably  chosen. 

A  function  which  is  developable  in  a  convergent  power  series  in  ascending 
positive  integral  powers  and  in  the  finite  portion  of  the  plane  nowhere 
becomes  infinite  or  indeterminate  is  an  integral  transcendental  function 
(see  Chapter  I).  Such  a  function  is  <j>(u)  above. 

The  question  is  asked :  Is  there  an  integral  transcendental  function  which 
has  besides  the  period  a  another  period  bf 

Liouville  [Crelle's  Journ.,  Bd.  88,  p.  277]  answered  this  question  by  prov 
ing  the  following  theorem:  An  integral  transcendental  function  which  is 
doubly  periodic  is  a  constant. 

We  need  only  study  the  function  within  the  first  or  initial  period  paral 
lelogram,  i.e.,  the  one  which  has  the  origin  as  a  vertex  and  which  lies  to  the 
right  of  this  vertex.  For  every  point  u  of  the  plane  is  congruent  to  a  point 
u'  in  the  first  parallelogram,  that  is, 

u  =  u'  +  ka  +  Ib, 

where  k  and  /  are  integers.  The  function  has  therefore  the  same  value  at 
u  and  at  u'. 

An  integral  transcendental  function  becomes  infinite  for  no  finite  value 
of  the  argument.  Consequently  the  function  remains  finite  in  the  first 
period-parallelogram  and  therefore  the  absolute  value  of  the  function  in 
this  parallelogram  is  smaller  than  a  certain  finite  quantity  M.  Further, 
since  the  function  at  points  without  the  first  period-parallelogram  always 
takes  such  values  as  it  has  within  this  parallelogram,  it  remains  in  the 

99 


100  THEOEY    OF   ELLIPTIC   FUNCTIONS. 

whole  plane  less  in  absolute  value  than  M  .     But  an  integral  transcendental 

function  x  9    ,•?•?, 

g(u)  =  a0  +  a>iu  +  a2u2  +  a3u3  +  •  •  • 

which  remains  finite  for  arbitrarily  large  values  of  u  is  a  constant,  since 
g(u)  can  remain  finite  only  if  &i  =  0  =  a2  =  #3  =  •  •  *     • 
The  following  is  a  more  direct  proof  of  Liouville's  Theorem. 

fc=+oo  2iti 

If  $(u)=   ^  Ake   a  U, 

the  condition  that  ^  +  &)  =  ^} 


/c=-oo 

and  consequently  &—  z 


/fc  —  6 

Since  -  is  an  irrational  quantity,  e    a     ^  1,  and  therefore 

Aft  =  0  (fc  =  ±  1,   ±  2,  .  .  .  ). 
It  follows  that 

&(U)    =   AQ] 

and  consequently  there  is  no  integral  transcendental  function  which  is 
doubly  periodic. 

ART.  84.  We  shall  now  seek  to  form  a  doubly  periodic  function  which 
has  the  character  of  a  rational  function  and  which  may  therefore  be  written 
in  the  form  \ 


where  &(u)   and  W(u)  are  integral  transcendental  functions.     We  may 
write  fc=+oo         2xi  /b=+<x>        k2™  u 

"'        ^^=          Bke    a  U> 


*<«>.  has 


k=  -oo  k=-<*> 

where  Ak  and  Bk  are  constants,  so  chosen  that  the  two  series  are  convergent. 

Since  $(w)  and  *F(u)  both  have  the  period  a,  their  quotient  <J>(u)  has  the 
period  a.     We  therefore  have  to  bring  it  about  that  the  quotient     ^  ' 
also  the  period  6. 

We  must  so  determine  the  functions  &(u)  and  W(u)  that 

&(u  +  b)=  T(u)  $(M), 
V(u  +6)=  T(u)V(u), 

where  77(w)  is  a  function  of  u.     If  we  succeed  in  this,  then 


or  ^(w)  has  also  the  period  6. 


CONSTRUCTION   OF  DOUBLY   PERIODIC   FUNCTIONS.       101 

It  will  be  advantageous  to  make  our  choice  so  that  <&(u  +  b)  has  the 
same  zero  as  <£(&),  and  consequently 


does  not  vanish  or  become  infinite  for  any  finite  value  of  u.     This  will  be 
effected  if  we  write 

T(u)  =  eG^u\ 

where  G(u)  is  an  integral  function  in  u. 

We  have  then  to  seek  a  function  4>w   and  a  function  "^u   so  that 


&(u  +  a)  =  <J>(iO,  ¥(M  +  a)  = 

$(M  +  6)=  e°M3>(u),        V(u  +  b)  = 

We  shall  next  bring  about  a  further  limitation  in  that  we  determine 
(tt)  and  "^(w)  so  that  G(u)  is  an  integral  function  of  the  first  degree  in  u. 
We  will  then  have 

4>O  +  a)=  <fr(w),  ¥(K  +  o)=  ¥(w), 

$(M  +  6) 


where  A  and  /*  are  constants  which  are  at  our  disposal.  We  shall  see  that 
there  is  an  infinite  number  of  such  functions. 

Hermite  *  called  them  "doubly  periodic  functions  of  the  third  sort 
(espece)." 

If  <£>(M  +  a)=  v$(u)  and  $(u  +  b)=  v'<E>(w),  where  v  and  i/  are  con 
stants,  one  or  both  being  different  from  unity,  then  &(u)  is  a  doubly  peri 
odic  function  of  the  second  sort;  and  if  v  =  1  =  i/  wre  have  the  doubly 
periodic  functions  of  the  first  sort,  which  are  properly  the  doubly  periodic 
functions. 

Note  that  the  wrord  sort  (espece)  used  here  in  no  manner  connects  a 
doubly  periodic  function  of  the  first  sort,  say,  with  an  elliptic  integral  of 
the  first  kind  (espece),  a  term  which  will  be  employed  later. 

*  Hermite  (Lettre  a  Jacobi;  Hermite's  (Euvres,  1.  1,  p.  18)  first  considered  these  func 
tions.  Briot  and  Bouquet,  Fonctions  Elliptiques,  p.  236,  called  them  "intermediary 
functions"  They  are  sometimes  called  quasi-  or  pseudo-periodic.  See  also  Hermite, 
"  Cours"  (4me  e"d.),  pp.  227-234;  Hermite,  Xote  sur  la  theorie  des  fonctions  in 
Lacroix,  Calcul  (6me  &L),  t.  2,  p.  384,  which  is  reprinted  in  Hermite's  (Euvres,  t.  2, 
p.  125;  Hermite,  Note  sur  la  theorie  des  fonctions  elliptiques,  Camb.  and  Dubl.  Math. 
Journ.,Vo\.  Ill  (1848);  Hermite,  CEhvres,  p.  75  of  Vol.  I;  Crelle,  Bd.  100;  Comptes 
Rendus  (1861),  t.  53,  pp.  214-228,  and  Comptes  Rendus  (1862),  t.  55,  pp.  11-18,  85-91; 
Biehler,  These,  1879;  Painleve,  Ann.  de  la  Faculte  des  Sciences  de  Toulouse,  1888; 
Appell,  Ann.  de  I'Ecole  Normale,  3d  Series,  Vols.  I,  II,  III  and  V;  Picard,  Comptes 
Rendus,  21  Mars,  1881.  The  Berlin  lectures  of  the  late  Prof.  L.  Fuchs  have  also  been 
of  service  in  the  preparation  of  this  Chapter. 


102  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  85.     From  the  formula 

k=  +00  k2ni 

$(w)  =   2)  Ake    a  ", 

k=  -oo 

it  follows  at  once  that 


.6 

If  with  Hermite  we  write  Q  =  e   °,  it  follows  that 

*=+»  k*™u 

&(u  +  b)=   2}  AkQ2ke    a     .  ,  (1) 

On  the  other  hand  we  had 

If  on  the  right-hand  side  we  write  for  3>(u)  its  value  and  put  A  =  =-^  g, 
we  have 

<b(u  +  b)=e^i\Ake£U(k+a)  (2) 

k=  —oo 

In  this  formula  k  is  an  integer  and  we  shall  choose  the  quantities  so 
that  g  is  also  an  integer.  2™ 

If  further  we  write  t  =  e  a  and  equate  like  powers  of  t  in  formulas  (1) 
and  (2),  we  have  for  the  determination  of  the  A' a  the  formula 

A      (~\2m —   pf-A 
slmty         —   &  -n-m-g- 

If  we  take  the  logarithms  of  both  sides  of  this  equation,  we  have 

/£  +  log  Am-g  =  log  Am  +  2  m  log  Q  +  n  2  ni,  (i) 

where  on  the  right  n  2  m  has  been  added,  since  the  logarithm  is  an  infinitely 
multiple-valued  function. 

We  shall  further  write  6 

.  o  *l  ~ v 

H  =  m  —  v,  so  that  e^  =  e   c     =  Q", 
a 

or  jj.  =  v  log  Q. 

Since  the  constant  /*  is  perfectly  arbitrary,  v  is  also  arbitrary.  It  follows 
directly  from  (i)  that 

=  2  m  —  v  + 


logQ  logQ 

We  note  that  m,  n,  and  g  are  integers,  and  we  seek  the  most  general 
solution  of  this  equation. 

If  for  brevity  we  put  — = — -  —  c*,  the  equation  (ii)  becomes 

Cm-g  -cm  =  2m-v  +  n  , — ^  .  (iii) 


CONSTRUCTION   OF   DOUBLY   PERIODIC    FUNCTIONS.       103 

To  determine  first  a  particular  solution  of  this  equation,  write 

ck  =  ak2  +  pk, 

where  the  constants  a  and  ft  are  to  be  determined. 
Since  cm-Q  =  a(m  —  g}2  +  3(m  -  g)     and 

cm  =  am2  4-  fim, 
we  have  from  equation  (iii) 

—  2  amg  +  ag2  —  pg  =  2  m  —  v  +  n  — '-^—- 

Since  this  equation  must  be  satisfied  for  every  value  of  m,  the  coefficients 
of  like  powers  of  m  on  either  side  of  it  must  be  equal.     We  thus  have 

—  2  ag  =2,     ag'2  —  t3g  =  —  v  + 
and  consequently 


We  may  give  to  the  arbitrary  constant  v  a  value  and  we  shall  write 
v  =  g.     It  follows  at  once  that 

1  -  n2-i 


These  values  written  in  the  formula 

ck  =  ak2  +  Pk 
will  give  the  particular  solution  of  the  equation 

cm-g  —  cm  =  2  m  —  v  +  n  - — '^—  •  (iii) 

We  may  write  the  general  solution  in  the  form 
cm  =  am2  +  pm  +  Cm, 

where  Cm  is  a  function  of  m. 

Writing  for  cm  its  value,  we  have 

l°%Am  =  am2  +  Pm  +  Cm,     or 

J        _  pam-  log  Q  +  (0m  +  Cw)  log  Q 
-*1  m         & 

Writing  for  a  its  value  from  above  and  putting  pm  +  Cm  =  Dm,  we  have 

=  Q~~°  eL 


104  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Finally,  putting  Dm  log  Q  =  log  Bm,  we  have 

_  m2 

Am  =  Q     °  Bm, 

where  Bm  is  a  new  function  of  m.  Here,  indeed,  we  have  not  deter 
mined  Am,  since  Bm  is  not  determined;  but  we  have  found  a  suitable  form 
for  Am. 

Returning  to  the  original  equation 

AmQ2m  =  e>*Am-g,     it  follows  that 

_  m_2  _  (TO-?)' 

Q2mQ     °Bm  =  Q°Q        *    Bm-0, 
or  Bm-g  =  Bm, 

where  m  and  g  are  integers. 

The  integer  g  being  arbitrary  we  shall  write  g  =  —  k,  where  &  is  a  positive 
integer.     We  thus  have 

Bm+k  =  Bm. 

It  follows  at  once  that 

Bk  =  B0, 
Bk+i  =  BI, 
Bk+2  =  B2, 

B2k-i  =  Bk-i, 

B'2k   =    Bjf   =    BQ. 

We  thus  see  that  the  constants  BQ,  BI,  B2,  .  .  .  ,  Bk-i  repeat  themselves 
but  are  otherwise  quite  arbitrary. 

It  has  thus  been  shown  that  the  function 


BmQke 


satisfies  the  functional  equations 

$(u  +  a)  = 


This  function  <&(u)  is  the  most  general  integral  transcendental  function 
which  satisfies  these  two  equations.  It  contains  the  k  arbitrary  constants 

~D  D  E>  O 

•DO,  o.it  k>2,   -   •   •   ,  *»*-!* 

ART.  86.  It  remains  to  be  proved  that  the  series  through  which  the 
function  4>(w)  has  been  expressed  is  convergent.  Instead  of  the  con 
vergence  of  the  series  itself,  we  may  consider  the  convergence  of  the  series 
of  moduli  of  the  single  terms,  that  is,  of  the  series 

m=  +00 

_    ^  w2      27ri       m 

*     Bm  Q  ¥  e~U     . 


CONSTRUCTION    OF   DOUBLY   PERIODIC    FUNCTIONS.        105 


In  this  series  the  coefficients  |  #0  |,  |  -#1  |  ,  •  •  •  ,  |  #&-i  |  repeat  them 
selves.  We  collect  all  those  terms  which  contain  |  B0  \  and  likewise  all 
those  which  contain  |  BI  |,  etc.,  and  take  |  B0  \,  \B'i\t  .  .  .  on  the  out 
side  of  the  summation  signs.  We  thus  distribute  the  above  series  into 


B 


If 


k  new  series  of  which  each  is  multiplied  by  one  of  the  quantities 
each  of  these  series  is  convergent,  then  the  product  of  each  one  of  them  by 
the  corresponding  |  B  \  is  convergent  and  therefore  also  the  sum  of  the 
products,  that  is,  the  above  series  of  moduli,  is  convergent.  If  this  series 
of  moduli  is  convergent,  it  follows  also  a  fortiori  that  the  series  which 
represents  $(u)  is  convergent. 

It  therefore  remains  to  prove  the  convergence  of  the  k  single  series. 
To  do  this  we  may  make  use  of  the  following  well-known  criterion  of  con 
vergence  : 

Suppose  we  have  given  a  series  composed  solely  of  positive  terms 

vi  +  v2  +  •   •   •  +  vm  +  -•-.. 

This  series  is  convergent  if  the  mth  root  of  the  mth  term,  that  is,  *\/^,  tends 
towards  a  dejinite  value  which  is  less  than  unity,  with  increasing  values  of  m. 
For  if  ^/vm  <  p  <  1,  then  is  vm  <  pm  <  1,  and  vm  +  1  <  pm+l  <  1,  etc., 
so  that  2vm  is  less  than  a  geometrical  series  in  which  p  <  1.  The  general 
term  in  the  above  series  is 


Q 
and  the  ?ftth  root  of  this  quantity  is 


Q 


2  « 


2  in 


The  second  of  the  above  factors  has  for  all  finite  values  of  u  a  definite 
value  which  is  independent  of  m.     For  the  other  factor  we  may  write 

Q 

If  we  put  -  =  a  +  ip  (where  p  ^  0,  since  -is  not  real),  we  have 
a  a 

-•b  _      , 

KI--KUZ-XP 


and 

It  follows  that 


.6 

TTt  —    I 

e   a  = 


-«*? 

becomes  arbitrarily  small 


If  p  is  a  positive  quantity,  the  quantity  e 
for  increasing  values  of  m,  which  proves  the  convergence  of  each  of  the 
above  series. 


106  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  condition  that  /?  be  positive  need  not  be  regarded  as  a  limitation. 
For  if  /?  is  negative,  we  form  the  quotient 


a    +  a     +  /          a 

where  the  coefficient  of  i  on  the  right  is  positive.     We  may  therefore  write 
|  =  a'  +  ;/?',  where  /?'  is  positive.     If  then  the  coefficient  of  i  in  -  is  nega 

tive,  we  interchange  b  and  a  in  the  whole  investigation  and  thus  form  a 
function  4>(w)  of  such  characteristics  that 


jri/fc 

'  a) 


The  function  $(u)  is  defined  by  the  series 


where  Q0  «  e^6. 

ART.  87.     If  k  =  1,  we  have  (Art.  85) 
$(w  +  a)  =  $(u), 

&(u  +  b)  =  e~"(2 
which  equations  are  satisfied  by  the  series 

m=+oo  m2        2n 


where 

In  this  case,  since  the  B's  are  all  equal,  we  may  write 


This  is  Hermite's  function  X(w),  when  we  make  B0  =  1.      It  is  the 
simplest  intermediary  function  and  is  called  the  Chi- function. 
For  k  =  2,  we  have 

$(u  +  a)  =  <£(w), 


(w  +  6)  =  6     a  (      h6) 


2  jri 


CONSTRUCTION   OF    DOUBLY   PERIODIC    FUNCTIONS.       107 

In  this  case  4>(w)  contains  the  two  arbitrary  constants  B0  and  BI.  We 
may  therefore  write 


where  $0(w)  =    2*  &**  °     '   '  (w  =  2  /i) 

fi=  -X 

,=  +*      (2,  +  l).2« 

$iOO  =  Q  e°  ,      (m  =  2a 


The  constants  B0  and  BI  being  arbitrary,  we  choose  B0  =  1,  and  BI  =  0, 
and  thus  have  a  particular  solution  4>o(w)  of  the  functional  equations; 
writing  BQ  =  0  and  BI  =  1  we  have  another  particular  solution  <&i(u). 

The  functions  ^oC^)  and  $i(t*)  are  the  remarkable  functions  first  intro 
duced  into  analysis  by  Jacobi  and  known  as  the  Jacobi  Theta-f unctions.* 
Jacobi  employed  a  somewhat  different  notation,  which  we  will  have,  if  we 
write 

.6 

Q2  =  er'l~a  =  q. 
It  follows  then  that 

^=  -foe  4  ;rt 

Y«2     a 


u=  —  X 


Jacobi  further  wrote  instead  of  a  the  quantity  4  K,  and  instead  of  b  the 
quantity  2  i'K',  and  consequently 


,-•*. 


The  above  functions  become 


«=  .a  «tu  2 

•too-  HIM*     ?1  2  ^2K 


*  In  his  memorial  address  Lejeune-Dirichlet  eulogized  Jacobi  as  follows  (see 
Jacobi,  Ges.  Werke,  I,  p.  14):  "Bedenkt  man,  dass  die  neue  Function  jetzt  das  panze 
Gebiet  der  elliptischen  Transcendenten  beherrscht,  dass  Jacobi  aus  ihren  Eigenschaften 
wichtige  Theoreme  der  hohreren  Arithmetik  abgeleitet  hat,  und  dass  sie  eine  wesent- 
liche  Rolle  in  vielen  Anwendungen  spielt,  von  welchen  hier  nur  die  vermittelst  dieser 
Transcendenten  gegebene  Darstellung  der  Rotationsbewegung  erwahnt  werden  mag, 
so  wird  man  dieser  Function  die  n-ichste  Stelle  nach  den  langst  in  die  Wissenschaft 
aufgenommenen  Elementartranscendenten  einraumen  miissen." 


108  THEOKY   OF   ELLIPTIC    FUNCTIONS. 

ART.  88.     If  we  put 


it  is  seen  that      <j>(u  +  a)  =  ^  +  a)  -  t 

a)      $ 


and 


6)         -** 


It  is  thus  shown  that  the  function  (f>(u)  is  a  doubly  periodic  function  hav 
ing  the  periods  a  and  6.     This  function  <f>(u)  cannot  be  a  constant,  for  if 


then  4>1(w)=  C<l>o  (w)>  which  is  not  true  since  $0(u)  is  developable  in  the 

2*t 

-  W 

even  powers  of  e  a      while  $i(iO  is  developable  in  the  odd  powers. 

The  functions  <£(w)  which  have  been  considered  do  not  become  infinite 
or  indeterminate  for  any  finite  value  of  u;  they  have  the  character  of 
integral  functions  and  may  be  developed  in  power  series  which  proceed  in 
positive  integral  powers.  They  are  integral  transcendental*  functions 
(Chapter  I). 

ART.  89.  Historical.  —  Abel  ((Euvres,  Sylow  and  Lie  edition,  T.  I,  p.  263 
and  p.  518,  1827-1830)  showed  that  the  elliptic  functions  considered  as  the 
inverse  of  the  elliptic  integrals  could  be  expressed  as  the  quotient  of  infinite 
products.  These  infinite  products  Jacobi  [Gesam.  Werke,  Bd.  I,  p.  198, 
1829]  introduced  into  analysis  under  the  name  of  Theta-f  unctions,  and  by 
expanding  them  in  infinite  series  (see  Chapter  X)  he  discovered  many  new 
properties  other  than  those  which  had  been  previously  employed  in  mathe 
matical  physics  by  French  mathematicians,  notably  by  Poisson  and  Fourier 
(Sur  la  Theorie  de  la  Chaleur). 

Jacobi  [Fund.  Nova,  p.  45;  Werke,  Bd.  I,  p.  497]  founded  the  whole 
theory  of  the  elliptic  functions  upon  these  new  transcendents,  which  made 
the  elliptic  functions  remarkably  simple,  as  well  as  their  application,  for 
example,  to  rotary  motion,  the  swing  of  the  pendulum,  and  innumerable 
problems  of  physics  and  mechanics;  also  through  these  Theta-f  unctions  the 
realms  of  geometry  were  essentially  widened  and  many  abstract  properties 
of  the  theory  of  numbers  were  revealed  in  a  new  light.  In  the  present 
treatise  these  Theta-functions  are  to  be  regarded  as  the  fundamental 
elements. 


CONSTRUCTION   OF   DOUBLY   PERIODIC   FUNCTIONS.       109 

ART.  90.    The  intermediary  functions  of  the  kth  order  or  degree.  —  It  is 
clear  that  we  may  write  the  function  4>(w)  of  Art.  85  in  the  form 


^ 

-      2, 


2k 
'      ' 


(J  =  0,  1,  .  .  .  ,  k  -  1). 

Such  functions,  for  reasons  given  in  Art.  92,  are  said  to  be  of  the  kth 
degree  or  order.  We  shall  next  prove  that  there  are  k  (and  not  more  than  k) 
independent  intermediary  functions  of  the  kth  order. 

Suppose  that  we  have  k  +  1  such  functions 


which  satisfy  the  functional  equations 

$(u  +  a)=  $(M), 

_  !Li^"(2 

3>(u  +  b}=e    a   ' 
These  functions  are  therefore  of  the  form 


(a  =  I,  2,  .  .  .   ,  k  +  1). 
We  have  at  once,  if  we  take  p(=l)  as  the  coefficient  of  ¥«(w), 

0  =  -  pVa(u)+  B0(a}3>0(u)+  Blw3>l(u)+  •  •  •  +B^k^(u) 
(a  =  1,  2,  .  .  .   ,  k  +  1). 

In  these  &  +  1  equations  we  may  consider  p,  <3>07  $1,  .  .  .  ,  ^-i  as 
unknown  quantities;  then,  since  the  equations  are  homogenous,  either 
their  determinant  must  be  zero,  or  all  the  unknown  quantities  are  zero. 
The  latter  cannot  be  the  case,  since  p  =  1. 

We  must  therefore  have 


D     (1) 
Dc\ 


p    (1) 

•  •  ,  Bk-i 

D   (21 


M),  B0 


(k+1 


=  0. 


If  this  determinant  is  expanded  with  reference  to  the  terms  of  the  first 
column,  we  have 


-  -  -  +  Ck+lVk+i(u)=  0, 
where  the  C"s  are  the  constant  minors  (sub-determinants). 

We  thus  see  that  there  exists  a  linear  homogeneous  equation  with  con 
stant  coefficients  among  any  k  +  1  intermediary  functions  of  the  kth 
degree. 


110  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

ART.  91.     The  zeros.  —  In  the  initial   period-parallelogram  there  is  a 
congruent  point  u'  corresponding  to  any  point  u  in  the  w-plane,  such  that 

u  =  u'  +  Xa  +  ffb, 
where  X  and  /*  are  integers. 
We  have 

&(u)  =  $>(u'  +  fib  +  Ja)  =  $(u'  +  /£&), 
and  further, 


~~('       ^ 


26)=  e*  ^(u  +  6), 


-  —  (2u+5b) 

a  ,$O  +  2  6), 


-  —  [2u+(2n-l)b] 

&(u  +  /£&)  =  e  $(u  +  («  -  1)  6). 

When  these  equations  are  multiplied  together,  we  have 


•  •  +2fi-l)] 

a 


or  >( 

It  follows  that 


Since  the  exponential  factor  is  different  from  zero,  it  follows  that 
can  only  vanish  when  &(u')  equals  zero.  We  may  therefore  limit  our 
selves  to  the  discussion  of  <£>(w)  within  the  initial  period-parallelo 
gram. 

Since  an  integral  transcendental  function  can  have  only  a  finite  number 
of  zeros*  (Art.  8)  within  a  finite  surface-area,  it  follows  that  there  are  only 
a  finite  number  of  zeros  of  <&(u)  within  the  period-parallelogram.  This 
parallelogram  may  be  constructed  in  different  ways.  If  from  any  point 
Q  in  the  w-plane  we  measure  off  both  in  length  and  direction  the  quantities 
a  and  6  and  draw  parallels  through  the  end-points,  we  have  a  period- 
parallelogram  of  the  function  with  the  periods  a  and  b.  If  starting  with 
this  parallelogram  we  cover  the  plane  with  similar  parallelograms,  it  is 
seen  that  the  plane  is  differently  divided  from  what  it  was  in  the  former 
distribution  of  parallelograms,  where  the  first  initial  parallelogram  had 
the  origin  as  one  of  the  vertices. 

*  Cf  .  Forsyth,  Theory  of  Functions,  p.  62. 


CONSTRUCTION   OF   DOUBLY   PERIODIC    FUNCTIONS.        Ill 


// 
L  —  L 


It  will  be  convenient  for  the  following  investigation  if  the  initial  period- 
parallelogram  is  so  situated  that  there  are  no  zeros  of  the  function  upon 
its  sides.  To  effect  this  let  QA'C'B'  be  any  period-parallelogram. 

As  there  can  be  only  a  finite  number  B 

of  zeros  of  4>(w)  within  this  parallelo-  B> 

gram,  it  is  evident  that  upon  the  line  /"" 

QBf  there  is  a  point  D  such  that  there  y  / 
is  no  zero  of  the  function  on  the  line 
DE  which  is  drawn  parallel  to  QA'  =  a. 
Similarly  there  will  be  a  point  F  on  the 
line  QAf  such  that  there  is  no  zero  of 
the  function  on  the  line  FG  drawn  parallel  to  QB'  =  b.  The  lines  DE 
and  FG  intersect  in  a  point  P,  say.  \Ve  take  P  as  the  vertex  of  a  new 
parallelogram  PACB.  We  shall  see  that  there  are  no  zeros  of  the  func 
tion  4>(w)  on  the  sides  of  this  parallelogram.  On  the  side  PE  there  is  by 
construction  no  zero.  Also  upon  EA  there  can  be  none  owing  to  the 
relation  &(u  +  a)  =  <b(u),  so  that  $(u)  takes  the  same  values  upon  EA 
as  upon  DP.  Upon  PG  likewise  by  construction  there  is  no  zero  of  the 
function  <b(u)  and  upon  GB  there  is  also  none,  since 


_ 
Fi    21 


(pu  +       =  e  u. 

Hence  upon  the  sides  PA  and  PB  there  are  no  zeros  of  the  function.  It 
follows  also  on  account  of  the  two  functional  equations  just  written  that 
there  are  no  zeros  on  the  sides  AC  and  BC. 

ART.  92.  We  may  now  apply  the  following  well-known  theorem  of 
Cauchy:*  If  a  function  3>(u)  within  a  definite  region,  boundaries  included, 
is  everywhere  one-valued,  finite  and  continuous,  and  if  N  denotes  the  number 
of  zeros  within  this  region,  then  is 


where  the  integration  is  to  be  taken  over  the 
boundaries  of  this  region  and  in  the  direc 
tion  such  that  the  region  is  always  to  the  left. 
This  theorem  is  applicable  to  our  func 
tion  &(u)  which  is  infinite  for  no  finite 
value  of  u.  The  region  in  question  is  the 


Fig.  22. 
period-parallelogram  PACB.    We  therefore  have ,  if  we  write  ty(u)  = 

2-iN  =   C    -fy(u)du+  f    ^r(u)du  +  \     ty(u}du  +  \      ^(u)d 
JPA  J AC  JCB  JBP 

*  Cf.  Forsyth,  loc.  cit.,  p.  63;  Osgood,  loc.  cit.,  p.  282.  Professor  Osgood  demands 
that  the  curve  be  analytic  (regular)  for  all  points  within  the  boundaries  and  continuous 
for  all  points  of  the  boundaries.  See  the  theorem  at  the  end  of  Art.  52. 


112  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  may  transform  these  integrals  of  the  complex  variable  into  integrals 
of  a  real  variable  t.  Let  u  take  the  value  p  at  P;  then,  since  PA  =  a,  we 
may  write  all  the  values  which  u  can  take  on  this  portion  of  line  PA  in  the 

form 

u  =  p  +  at, 

where  0  =  £  =  1. 
It  follows  that 


/     i!r(u)du=  a  I 
JPA  Jo 


+  at)dt. 

Further,  the  variable  u  has  at  A  the  value  p  +  a,  and  since  AC  =  b,  we 
have 


I     ^r(u}du  =  b  I 
JAC  Jo 


a  +  bt)dt. 

Similarly  u  has  at  B  the  value  p  +  b,  and  therefore  all  values  of  ^  on  CB 
have  the  form  p  +  b  +  at,  and  consequently 

r  r°  ci 

I     yr(u}du  =  a  I    "fy(p  +  b  +  at)dt  =  —  a  I    ty(p  +  b  +  at)dt. 

J AC  Jl  Jo 

Finally  we  have  in  the  same  manner 

r  r°  r1 

I     ifr(u)du  =  b  I    y(p  +  bt)dt=  —  I    y(p  H-  bt)dt. 

J  BP  **\  JQ 

It  is  thus  seen  that 

2  niN  = 

fir  -|  rir  n 

a  I    \'^r(p  +  at)—'^r(p  +  b+at)\dt  +  bl    \*Y(P  +  a  +  bt)  —  ^(p  +  bt)  \dt. 
Jo  L  J          «^o  L  J 

Further,    since    $0  +  a)  =  $(u)    arid    3>(u  +  b)  =  e     a  &(u),    it 

follows  at  once  through  logarithmic  differentiation  that 

^r(u  +  .a)  =^r(u)     and     ijr(u  +  b)  =^(u) • 

a 

These  values  substituted  in  the  above  integrals  give 


F0      a 

or  .V  =  k. 

We  thus  see*  that  the  number  of  zeros  of  the  function  4>(w)  which  lie 
within  the  period-parallelogram  is  equal  to  the  integer  k  which  appears 
in  the  second  functional  equation  which  <f>(w)  satisfies. 

*  Cf.  Hermite,  "  Cours"  [4th  ed.],  p.  224. 


CONSTRUCTION   OF   DOUBLY   PERIODIC   FUNCTIONS.       113 

In  algebra  we  say  an  integral  rational  function  which  vanishes  for  k 
values  of  u  in  the  w-plane  is  of  the  kth  degree.  In  a  corresponding  manner 
we  say  of  our  function  $(u),  it  is  of  the  kth  degree  or  order,  because  it 
vanishes  at  k  points  within  the  period-parallelogram. 

ART.  93.     For  k  =  1,  we  had  in  Art.  87 

&(u  +  a)  =  3>(u), 

--(2u+6) 

$>(u  +  6)  =  e  $>(u). 

After  the  theorem  just  proved  we  know  that  there  is  one  and  only 
one  zero  of  the  function  <b(u)  which  satisfies  these  two  functional  equations 
in  the  period-parallelogram.  We  shall  seek  this  zero  in  the  initial  period- 
parallelogram.  We  had 

m= +x  2jrt' 


m=  —30 

Writing  m  =  —  (n  -f  1)  in  this  formula,  it  becomes 


2  ri 
ot 


Qn*e 

n=  -x 

n=+x          ^(2n6+6-2nu-2t*) 

Tea 


If  we  give  to  u  the  value  — —  in  the  above  formula,  it  becomes 


«[n6_(n+i)0] 

Tea 


n=+oo 

Qw2+n(-l)n. 

n= —x  n= —x 

If  we  also  write  a          in  the  original  expression  for  X(w),  it  becomes 


Comparing  the  two  expressions  thus  obtained  for  Xj— ~ — J,  it  is  seen  that 

they  differ  from  each  other  only  in  sign,  and  consequently  it  necessarily 
follows  that 


114  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Since  the  zero  of  the  intermediary  function  <b(u)  of  the  first  order,  i.e.,  of 
X(w),  is  the  intersection  of  the  diagonals  of  the  initial  period-parallelogram, 
it  follows  that  X(w)  =  0  at  all  the  intersections  of  the  diagonals  of  the 
parallelograms  which  are  congruent  to  this  initial  parallelogram. 

Remark.  —  The  question  might  be  raised  as  to  whether  there  were  zeros 
of  X(w)  on  the  boundaries  of  the  initial  period-parallelogram.  We  saw 
in  Art.  91  that  it  was  always  possible  so  to  place  the  period-parallel 
ogram  that  its  boundaries  were  free  from  zeros.  If,  however,  we  con 
sider  as  we  do  here  a  definite  period-parallelogram,  viz,,  the  one  where  the 
origin  is  the  vertex  and  which  lies  to  the  right  of  the  origin,  we  do  not 
a  priori  know  that  there  is  no  zero  of  X(w)  upon  its  boundaries. 

Suppose  that  the  period-parallelogram  which  has  u  =  p  as  one  of  its 
vertices  is  so  drawn  that  there  are  no  zeros  upon  its  boundaries.  There 
c  is  one  zero  within  the  period-parallelogram,  since 
<&(u)  is  of  the  first  degree.  The  value  of  u  at  this 
point  may  be  expressed  in  the  form 

p  +  ha  +  vb, 

j£    23  where  A  and  v  are  proper  fractions.     If  now  we 

cover  the  w-plane  with  congruent  parallelograms, 
there  does  not  lie  a  zero  of  X(tt)  on  any  of  the  boundaries  of  these  paral 
lelograms,  and  within  each  parallelogram  there  is  always  one  and  only 
one  zeroo  Since  all  the  zeros  are  congruent  one  to  the  other  and  since 

from  above  is  one  of  them,  we  must  have 


where  g  and  I  are  integers.  Every  zero  of  X(w)  may  be  expressed  in  this 
form,  and  therefore  also  the  zero  which  we  suppose  may  lie  upon  one  of 
the  boundaries  of  the  initial  period-parallelogram,  L 

say  at  L,  where 

L  =  b  +  da, 

$  being  a  proper  fraction. 
We  would  then  have 


a 
ga  +  lb  =  b  +  $a,  Fig.  24. 


and  consequently 


b  1  - 2#  +  2# 

But  the  right-hand  side  of  this  expression  is  a  rational  number,  which  is 
contrary  to  what  has  been  proved  in  Art.  71.  When  L  lies  upon  any  other 
side  of  the  parallelogram,  we  may  derive  a  similar  result  and  thus  by  a 
reductio  ad  absurdum  show  that  there  does  not  lie  a  zero  of  X(w)  upon  the 
boundary  of  the  initial  period-parallelogram. 


CONSTKUCTION   OF   DOUBLY   PEEIODIC   FUNCTIONS.       115 

THE  GENERAL  DOUBLY  PERIODIC  FUNCTION  EXPRESSED  THROUGH  A 
SIMPLE  TRANSCENDENT. 

ART.  94.  We  shall  next  consider  a  doubly  periodic  function  F(u) 
which  has  nowhere  in  the  finite  portion  of  the  plane  an  essential  singularity. 
Such  a  function  has  only  a  finite  number  of  zeros  and  a  finite  number  of 
infinities  within  a  finite  area.  We  may  limit  our  study,  as  shown  above, 
to  the  initial  period-parallelogram.  We  shall  assume  that  within  this 
parallelogram  the  function  F(u)  has  the  infinities  u\,  u^  .  .  •  ,  un]  and 
we  shall  further  assume  that  these  infinities  are  of  the  first  order,  so  that 
in  the  neighborhood  of  any  one  of  them,  u\  say,  F(u)  has  the  form 

F(u)  =  — - —  +  c0  +  CI(M  -  MI)  +  c2(u  -Mi)2  +  •  •  •  , 

u  —  u\ 

where  d  and  the  c's  are  constants. 

We  shall  see  that  every  such  function  may  be  expressed  through  the 
general  intermediary  functions  3>(u).  We  shall  form  such  a  function 
where  the  integer  k  is  taken  equal  to  n  -f  1  and  which  therefore  satisfies 
the  two  functional  equations 

$(M  +  a)  =  $(M), 

There  being  n  +  1  arbitrary  constants  in  this  function,  we  may  write  it 
in  the  form 


The  constants  BQ,  B\,  .  .  .  ,  Bn  may  be  so  determined  that  the  function 
$(u)  becomes  zero  of  the  first  order  on  the  points  HI,  u2,  .  .  .  ,   un. 
For  write 

=  0, 

-  0, 


In  these  equations  we  may  consider  the  B's  as  the  unknown  quantities. 
We  have  then  n  equations  with  n  +  1  unknown  quantities,  from  which 
we  may  determine  the  ratios  of  the  J5's  so  that  <f>(w)  becomes  zero  of  the 
first  order  at  all  the  points  HI,  u2,  .  .  .  ,  un.  By  hypothesis  F(u)  be 
comes  infinite  of  the  first  order  on  all  these  points. 
Form  the  product 

/(*)-*(*)*•<*). 


116  THEOKY   OF   ELLIPTIC    FUNCTIONS. 

It  is  seen  that 

<j>(u  +  a)  =  $(u  +  a)  F(u  +  a)  =  <f>(w)  F(u)  =f(u); 
and  also 

f(u  +  b)  =  &(u  +  b)F(u  +  b) 


=  e  $(tt)  F(u), 

-(n  +  l)-(2w  +  6) 

or  f(u  +  b)  =  e  f(u}. 

From  this  it  is  seen  that  f(u)  is  also  one  of  the  intermediary  functions 
which  satisfies  the  same  functional  equations  as  does  $(u).  Further, 
since  $(u)  becomes  zero  of  the  first  order  at  the  same  points  at  which 
F(u)  is  infinite  of  the  first  order,  the  product  f(u)  =  &(u)  F(u)  is  nowhere 
infinite  in  the  finite  portion  of  the  plane.  A  one-valued  analytic  function 
which  does  not  have  an  essential  singularity  in  the  finite  portion  of  the 
plane  and  in  this  portion  of  plane  is  nowhere  infinite,  is  an  integral  tran 
scendental  function;  and,  as  there  are  only  n  +  1  such  functions  that  are 
linearly  independent  (cf.  Art.  90),  it  follows  that 

f(u)   =  C03>o(u)   +  Ci$i(w)   +  C2$2(U)   +  •     •     •   +  Cn$n(u), 

where  the  C"s  are  constant. 

It  is  also  seen  that  ff».\ 

F(u)  =  £&  . 
*(«) 

We  consequently  have  the  theorem:  Any  arbitrary  doubly  periodic 
function  which  has  only  infinities  of  the  first  order  may  be  expressed  as  the 
quotient  of  two  integral  transcendental  functions,  both  of  which  satisfy  the 
same  functional  equations. 

ART.  95.  By  means  of  the  X  (it)  -function  we  can  make  the  above 
theorem  more  general  in  that  the  order  of  the  infinities  of  F(u)  is  not 
restricted. 

We  have  noted  in  Art.  93  that  X(u)  is  zero  for  the  value  u  =  ?L±J!  =  Cf 
say.  Hence  X(u  +  c)  =  0  for  u  =  ha  (X  =  0,  1,  2,  .  .  .  ). 

If  we  write  X(u  +  c)=Xi(u),  it  is  seen  that  Xi(u)  =  0  for  u  =  0.  We 
also  observe  that  the  function  Xi(u)  satisfies  the  two  functional  equations 

Xi(u-+a)  =Xl(X>, 


Xi(w  +  b)  =  e    '  XI(M). 

We  have  immediately  the  following  relations: 

7T"l 

-  —  (2u-2u2+a  +  b  +  b) 

Xi(u  —  u2  +  6),  =  e  Xi(u  —  u2), 


—  Uk  +  6)  =  e 


CONSTRUCTION   OF   DOUI3LY   PERIODIC    FUNCTIONS.        117 


If  we  put  V(u)  =  XI(M  -  MI)  XI(M  —  u2)  .  .  .  XI(M  —  Uk),  it  is  seen 
that 

V(u  +  a)  =  ¥(w), 


-fc  —  (2u+6) 

=  e      a 


provided  that 

A;  (a  +  6)  -  2(ui  +  u2  4-  •   •   •  +  uk)  =  2  ma; 
that  is,  if  iii  -f  u2  +  •   •   •  +Uk  =  kc  —  ma,  (1) 

where  m  is  any  integer. 

Hence  if  k  =  n  +  1,  the  function  ^f(u)  becomes  zero  on  any  n  arbitrary 
points  ui,  U2,  .  .  .  ,  un,  while  the  other  zero  must  satisfy  equation  (1). 
As  some  of  the  points  u\,  u2,  .  .  .  ,  un  may  be  made  equal  to  one  another, 
it  is  seen  that  the  zeros  are  not  restricted  to  being  of  the  first  order  in 
¥(M).  We  may  therefore  let  ¥  (u)  take  the  place  of  f(u)  in  the  preceding 
article  and  mutatis  mutandis  have  the  same  result  as  stated  there. 

ART.  96.  It  is  convenient  to  form  here  a  function  which  becomes 
infinite  of  the  first  order  for  u  =  0,  u  =  a,  u  =  2  a,  •  •  •  .  Such  a  func 
tion  is  the  Zlta-f  unction  (see  Art.  97), 


This  function  ZQ(U)  is  one-valued  in  the  entire  M-plane  and  has  an  essen 
tial  singularity  only  at  infinity.  By  means  of  this  fundamental  element 
Her  mite*  has  given  a  general  method  of  expressing  any  one-valued  doubly 
periodic  function  which  in  the  finite  portion  of  the  plane  has  no  essential 
singularity. 

We  shall  so  choose  the  period-parallelogram  that  F(u)  does  not  become 
infinite  on  its  boundaries.  If  the  function  F(u)  is  infinite  of  the  /Ith 
order  say  at  MI,  the  development  in  the  neighborhood  of  this  point  is 

F(u)  -         6*         +         **-*         +  -   -  .  +  -^-  +  P(u  -  1*0, 

^  ~ 


the  6's  being  constants. 

We  shall  now  give  a  method  of  representing  this  function  when  for 
every  infinity  the  complex  of  all  the  negative  powers  is  known.  This 
complex  of  negative  powers  we  have  called  (in  Chapter  I)  the  principal 
part  of  the  function.  We  introduce  a  new  variable  £  and  form 


where  now  u  is  to  play  the  role  of  a  parameter,  being  a  point  within  the 
initial  parallelogram,  while  c  is  the  variable.     We  consider  in  the  c-plane 

*  Hermite,  Ann.  de  Toulouse,  t.  2  (1888),  pp.  1-12,  and  "Cours"  (4th  ed.),  p.  226. 


118  THEORY   OF   ELLIPTIC   FUNCTIONS. 

a  period-parallelogram  of  F(£),  upon  whose  boundaries  there  is  no  infinity 
of  F(£). 

The  function  /(£)  becomes  infinite  within  this  period-parallelogram  on 
the  points  ui,  u2,  .  .  .  ,  un,  the  points  at  which  F(g)  is  by  hypothesis 
infinite;  and  /(£)  is  infinite  also  at  the  additional  point  £  =  u,  since 
Zq(0)'-ao. 

We  form  the  sum  of  the  residues  of  /(£)  with  regard  to  all  the  above 
infinities  and  have  after  Cauchy's  Residue  Theorem 


where  the  integration  is  to  be  taken  over  the  sides  of  the  period-parallel 
ogram  and  in  such  a  way  that  the  surface  of  the  parallelogram  is  always 
to  the  left.  We  therefore  have 


a 


.  fp+a  fp+a- 

,.  ,T  ...twites /(?)  =  /    m<K+  I    /(« 

//  //  *s v  *Jp+a 

/ 


Fig.  25.  or,  as  in  Art.  92, 

£  5)  Res/(£)  =  a  \  f(p  +  at)dt  +  b  \ 

-  a  \  f(p  +  6  +  at)dt  -  b  \  f  (p  +  bt)dt. 
Jo  Jo 


Further,  since    ZQ(V  +  a)  =  Zo(v)| 
Z0(V  +  6) 


it  follows  that 

/(£  +  a)  =  F(^  +  a)  Z0(^  -  <f  -  a)  = 

and  /(£  +  6)  =  F(^)  <  Z0(^  -  0  +  -^  |  =/(c) 

We  therefore  have 


o 


and  consequently  * 


There  is  no  infinity  of  the  function  F  on  the  path  of  integration,  this 
being  a  side  of  the  parallelogram  above.     Hence  the  integral  on  the  right 

*  Cf.  Hermite,  loc.  cit.,  p.  226. 


COXSTKUCTION   OF   DOUBLY   PEKIODIC   FUNCTIONS.       119 

has  a  definite  value,  a  value  which  is  independent  of  u,  and  as  it  does  not 
contain  f,  it  is  a  definite  constant. 

ART.  97.  We  shall  next  determine  by  direct  computation  the  sum 
of  the  above  residues  of  /(£). 

We  had  -,,     , 


X(l7  +  C) 

The  function  X(y  +  c)  becomes  zero  of  the  first  order  for  v  =  0,  and  is 
one-valued  and  finite  for  all  finite  values  of  v. 
Its  development  is  therefore  of  the  form 

X(v  +  c)  =  riv  +  /-2v2  +  •  •  •  , 

where  the  fs  are  constant  and  f\  ^  0. 
Through  differentiation  it  is  seen  that 

X'O  +  c)  =  fi  +  2  ?2v  +  •   •   •  , 
and  consequently 


We  note  that  the  residue  of  Z(v)  with  respect  to  v  =  0  is  unity.  This 
function,  as  shown  in  the  sequel,  has  in  regard  to  the  doubly  periodic 
functions  the  same  relation  as  has  cot  u  with  respect  to  the  simply  periodic 

functions  and  as  has  -  to  the  rational  functions. 
v 

If  for  v  we  substitute  u  —  c,  we  have 


c  —  u 


which  is  the  development  of  Z0(u  —  c)  in  the  neighborhood  of  £  =  u. 

We  next  form  the  corresponding  development  of  F(£).  In  the  interior 
of  the  period-parallelogram  the  function  F(£)  becomes  infinite  at  the 
points  HI,  u2,  .  .  .  ,  UH  but  not  at  u.  Hence  we  may  develop  F(£)  by 
Talor's  Theorem  in  the  form 


Further,  since  /(£)=  ^(f)  Z0(u  -  c),  it  follows  that 


and  consequently  Res  /(f)  =  _  F(u). 

^  =  U 

We  saw  above  that  2  Res/($)  is  independent  of  u,  but  as  shown  here, 
the  single  residues  are  dependent  upon  this  quantity. 


120  THEORY    OF   ELLIPTIC    FUNCTIONS. 

ART.  98.  We  shall  next  calculate  the  residues  of  /(£)  with  respect  to 
the  other  infinities  u\,  u^,  .  .  .  ,  un.  Suppose  that  the  function  F(£) 
becomes  infinite  of  the  ^th  order  on  the  point  u\,  so  that  F(£)  when  ex 
panded  in  the  neighborhood  of  this  point  is  of  the  form 


where  the  6's  and  c's  are  constants. 

For  the  value  £  =  u\  the  function  Z0(u  —  £)  is  not  infinite  and  may 
be  developed  by  Taylor's  Theorem  in  the  form 


It  follows  that  the  coefficient  of  -  -  in  the  product  F(£)Zo(w  —  £)  is 


, 

which  is  the  residue  of  /(£)  with  respect  to  the  infinity  £  =  u\.  The  resi 
dues  with  respect  to  the  other  infinities  u2,  u3,  .  .  .  ,  un  are  found  in  the 
same  manner.  The  6's  and  X,  of  course,  have  different  values  for  each 
of  these  points. 

Let  the  orders  of  infinity  at  Uk  be  fa  (k  =  1,  2,  .  .  .  ,  ri)  and  in  the 
neighborhood  of  the  infinity  Uk  let  the  principal   part  of  the  function 


(U  —  Uk)^k         (U  —  Uk)Kk~l          (U  —  UkYk~2  U  —  Uk 

It  follows  at  once  that 

k  =  n  p  . 

V     Res  m  -  5)  1  4k,  iZo(«  -  uk)  -^ 
l^i,  fcTi  L 


We  also  saw  that  Res  /(c)  =  —  F(w),  which  must  be  added  to  the  sum 
just  written. 

On  the  other  hand  we  had 

Res/(£)  =  -  /    F(p  +  at)dt  =  C,  say, 

«/o 
where  C  is  a  constant. 

Equating  these  two  expressions  for  the  sum  of  the  residues,  we  have 
F(u)=C+  j$\bktlZ0(u-uk)-  ^fz0'(u-Uk)  +  b-Mz0"(u-Uk)->  -  - 


which  is  the  required  representation  of  the  doubly  periodic  function  F(u). 
We  thus  see  that  a  doubly  periodic  function  may  be  expressed  through  a 
finite  sum  of  terms  that  are  formed  of  the  function  Z0  and  its  derivatives. 


CONSTRUCTION    OF   DOUBLY   PERIODIC    FUNCTIONS.        121 

EXAMPLE 

Show  that  two  doubly  periodic  functions  with  the  same  periods  and  the  same 
principal  parts  differ  only  by  an  additive  constant. 

In  Chapter  XX,  several  methods  of  representing  a  doubly  periodic 
function  will  be  found  and  the  consequences  which  result  therefrom  will 
be  derived.  All  these  methods,  however,  are  little  other  than  different 
interpretations  of  the  above  formula. 

It  is  seen  at  once  from  this  formula  that  we  may  represent  a  doubly 
periodic  function  when  its  principal  parts  are  given,  the  function  being 
completely  determined  except  as  to  an  additive  constant.  This  expres 
sion  for  a  doubly  periodic  function  is  the  analogue  of  the  formula  for  the 
decomposition  of  a  rational  function  into  its  simple  fractions  or  of  the 
decomposition  of  a  simply  periodic  function  into  its  simple  elements  (see 
Arts.  11  and  25).  It  may  be  shown  that  the  latter  cases  may  be  derived 
from  the  former  by  making  one  of  the  periods  infinite  for  the  case  of  the 
simply  periodic  functions,  and  by  causing  them  both  to  be  infinite  for  the 
rational  functions. 

ART.  99.  There  is  a  restriction  with  respect  to  the  constants  that 
appear  in  the  above  development. 

We  saw  that 


Z0(v  +  a)  =  Z0(r)     and     Z0(r  +  b)  =  Z0(r  )  - 


a 


It  follows  that  Z0(v)  is  not  a  doubly  periodic  function;  but  all  its  derivatives 
are  doubly  periodic,  since  we  have 

Z0'(v  +  a)  =  Z0'(t>), 
Z0'(v  +  b)  =  Z0'(i>),  etc. 

Hence  under  the  summation  sign  of  the  preceding  article  all  terms  except 
the  first  are  doubly  periodic.  Further,  since  F(u  +  6)  =  F(u),  it  also 
follows  that 


Since  Z0(u  -  uk  +  6)  =  Z0(u  -  uk)  -  —  > 

a 

it  is  evident  from  the  equality  of  the  two  summations  just  written  that 

k=n 

i-0,     or          6,,!  =  0. 


k=l  k=l 


We  thus  have  the  very  important  theorem:  The  sum  of  the  residues  within 
a  period-parallelogram  of  a  doubly  periodic  function  with  respect  to  all  of  its 
infinities,  is  equal  to  zero. 


122 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


If  we  wish  to  form  a  doubly  periodic  function,  when  its  principal  parts 
with  reference  to  its  infinities  are  given,  the  restriction  just  mentioned 
must  be  imposed  upon  the  constants. 

ART.  100.     We  may  prove  in  a  different  manner  that 

2  ResF(u)  =  0. 

Take  any  period-parallelogram,  upon  the 
sides  of  which  there  are  no  infinities  of  F(u). 


Fig.  26. 


/f        Then  by  Cauchy's  Residue  Theorem 
2  m  V  Res  F(u)  =  (    F(u)du. 

^  JvACB 


But  from  Art.  92  we  have 


I      F(u)du  =  a  I   F(p  +  at)dt  +  b  \F(p  +  a  +  bt)dt 

JpACB  i/O  t/0 

b  +  at)dt  -  b 


Further,  since 
it  follows  that 


F(u  +  a)  =  F(u)  =  F(u  +  b), 
2  Res  F(u)  =  0. 


ART.  101.  It  follows  directly  from  the  above  representation  of  a  doubly 
periodic  function  that  it  cannot  be  an  integral  transcendental  function 
(cf.  Art.  83).  In  this  case  all  the  quantities  bk,it  &*,2,  •  •  •  ,  bk,  A*  would 
be  zero  and  consequently 

F(u)  =  C. 

It  also  follows  that  a  doubly  periodic  function  cannot  be  infinite  of  the 
first  order  at  only  one  point  of  the  period-parallelogram.  For  if  u\  were 
such  a  point,  then  is 

F(u)= 


in  the  neighborhood  of  this  point,  and  consequently 

SResF(^)  =  6ltl. 

But  as  the  sum  of  the  residues  is  equal  to  zero,  it  also  follows  that  61,1  =  0 
and  consequently  F(u)  would  be  an  integral  transcendent.  But  an  inte 
gral  transcendental  function  with  two  periods  is  a  constant  (Art.  83).  We 
have  consequently  the  following  theorem  due  to  Liouville:  A  doubly 
periodic  function  must  have  at  least  two  infinities  of  the  first  order  within 
the  period-parallelogram,  or  it  must  be  infinite  of  at  least  the  second  order 
on  one  such  point. 


CONSTRUCTION   OF   DOUBLY   PERIODIC   FUNCTIONS.       123 

ART.  102.  We  have  then  two  different  methods  which  may  be  followed 
in  the  treatment  of  the  doubly  periodic  functions,  the  one  where  the  two 
infinities  of  the  first  order  in  the  period-parallelogram  are  distinct,  which 
is  the  older  method  employed  by  Jacobi,  say  z  =  snu;  while  the  other 
method  where  the  function  becomes  infinite  of  the  second  order  is  the  one 
followed  by  Weierstrass,  and  in  this  case  z  considered  as  a  function  of 
u  is  written  z  =  $>u.  The  notation  in  the  two  different  cases  is  inserted 
here,  as  it  is  convenient  to  refer  to  the  two  methods  by  means  of  this 
notation  before  the  general  treatment  of  these  particular  functions  is 
considered. 

In  the  next  Chapter  it  will  be  shown  that  a  doubly  periodic  function 
which  becomes  infinite  at  n  points  (the  order  being  finite  at  each  point)  is 
algebraically  expressible  through  either  one  of  the  above  simple  forms 
z  =  snu  or  z  =  <#u\  and  consequently  the  general  theory  of  doubly 
periodic  functions  is  reduced  to  the  consideration  of  the  two  simpler 
cases. 

THE  ELIMINANT  EQUATION. 

ART.  103.  We  have  shown  in  Chapter  III  that  a  one-valued  simply 
periodic  function  which  in  the  finite  portion  of  the  plane  has  no  essential 
singularity  and  which  takes  within  a  period-strip  any  value  only  a  finite 
number  of  times,  satisfies  an  algebraic  differential  equation  in  which  the 
independent  variable  u  does  not  explicitly  enter.  In  Chapter  II  we  have 
seen  that  associated  with  every  one-valued  analytic  function  which  has 
an  algebraic  addition-theorem  there  exists  an  equation  of  the  form  just 
mentioned.  W^e  shall  see  later  in  Art.  158  that  every  one-valued  doubly 
periodic  function  has  an  algebraic  addition-theorem,  so  that  (see  Art.  35) 
the  notion  of  the  doubly  periodic  function  and  of  the  eliminant  equation 
is  seen  to  be  coextensive  for  the  one-valued  functions. 

We  wish  now  to  show  that  there  is  an  eliminant  equation  which  is 
associated  with  every  one-valued  doubly  periodic  function.  First,  how 
ever,  it  is  necessary  to  consider  certain  preliminary  investigations. 

ART.  104.  Suppose  that  the  doubly  periodic  function  F(u)  has  n 
infinities  of  the  first  order  within  a  period-parallelogram,  or  if  it  becomes 
infinite  of  the  ^th  order  on  any  point,  let  this  point  be  counted  as  ^  infin 
ities  of  the  first  order,  so  that  the  totality  of  infinities  is  still  n.  Let  v  be 
any  arbitrary  quantity  and  consider  the  number  of  solutions  of  the  equa 
tion 

F(u)  =  v 
within  a  period-parallelogram. 

After  the  same  method  by  which  we  constructed  a  period-parallelogram 
which  had  no  infinities  upon  its  boundaries  we  may  also  construct  one 
which  has  no  zero  of  the  function  F(u)  —  v  upon  the  boundaries.  We 


124  THEORY   OF   ELLIPTIC    FUNCTIONS. 

may  therefore  assume  that  there  are  no  zeros  or  infinities  of  the  function 
F(u)  —  v  upon  the  boundaries  of  our  period-parallelogram. 
Consider  next  the  function 


=  F(u)-v. 

It  is  a  doubly  periodic  function  with  the  same  periods  as  F(u),  viz.,  a 
and  b.  As  it  becomes  infinite  at  the  same  points  as  F(u),  it  has  n  infinities 
within  the  period-parallelogram. 

Form  next  the  logarithmic  derivative  of  G(u), 

G'(u] 


= 

The  function  H(u)  has  the  periods.  a  and  b  and  becomes  infinite  at  the 
points  where  G'(u)  is  infinite  and  also  where  G(u)  is  zero. 
Let  u\  be  an  infinity  of  G(u)  of  the  Ath  order,  so  that 

G(u)  =  (u  -  UI)~*GI(U),     where  GI(UI)  ^  0. 
We  then  have  (Art.  4)  in  the  neighborhood  of  HI, 

H(u)  =  --  -  —  +  P(u  -  m), 

u  —  u\ 

so  that 

ResH(u)  =-  I, 

u=u\ 

that  is,  the  residue  of  H(u)  with  respect  to  u\  is  the  order  of  the  infinity 
of  G(u)  at  the  point  u\  with  the  negative  sign. 

Suppose  next  that  w\  is  a  zero  of  G(u)  of  the  jj.  th  order,  so  that 

G(u)  =  (u  —  wi)nG2(u),     where  G2(wi)  ^  0. 
We  then  have  in  the  neighborhood  of  w\ 


H(u)  =  —  —  +  P(u  -  wi),     or 
u  —  wi 

Res#(V)  =  /£, 

U  =  Wi 

that  is,  the  residue  of  H(u)  with  respect  to  a  zero  of  G(u)  is  equal  to  the  order 
of  the  zero  at  this  point. 

Further,  since  the  sum  of  the  residues  of  a  doubly  periodic  function  with 
respect  to  all  its  infinities  within  a  period-parallelogram  is  zero,  it  follows 

that 

-  Z/t  +  S/*  =  0, 

where  SA  denotes  the  sum  of  the  infinities  of  the  function  G(u)  in  a  period- 
parallelogram,  an  infinity  of  the  Ath  order  counting  as  A  simple  infinities, 
and  where  H/z  denotes  the  number  of  zeros  of  the  first  order  of  G(u),  a 


CONSTRUCTION   OF   DOUBLY   PERIODIC   FUNCTIONS.       125 

zero  of  the  /*th  order  counting  as  /*  zeros  of  the  first  order.  Since  G(u)  = 
F(u)  —  v,  it  follows  that  the  number  of  roots  of  the  equation 

F(u)  -v  =  0 

within  a  period-parallelogram  is  equal  to  the  number  of  infinities  of  the 
first  order  of  the  function  F(u)  within  this  parallelogram. 

It  follows  that  a  doubly  periodic  function  F(u)  takes  within  every  period- 
parallelogram  any  value  v  as  often  as  it  becomes  infinite  of  the  first  order 
within  this  period^parallelogram* 

ART.  105.  Let  z  =  F(u)  be  a  doubly  periodic  function  of  the  nth 
order  with  the  primitive  periods  a  and  b  and  let  w  =  G(u)  be  a  doubly 
periodic  function  of  the  fcth  order  with  the  same  periods.  Neither  of  these 
functions  is  supposed  to  have  an  essential  singularity  in  the  finite  portion 
of  the  u-plane.  We  assert  that  there  exists  an  algebraic  equation  with 
constant  coefficients  connecting  z  and  w. 

For  if  a  definite  value  is  given  to  z  there  are  n  values  of  u,  say  ui,  u2, 
.  .  .  ,  un,  for  which  F(u)  =  z.  If  we  write  these  values  of  u  in  w  —  G(u), 
we  have  n  values  of  G(u),  say  w\  =  G(UI),  w2  =  G(u2),  .  .  .  ,wn  =  G(un). 
Hence  the  variable  z  is  related  to  the  variable  w  in  such  a  way  that  to 
one  value  of  z  there  correspond  n  values  of  w  and  similarly  to  one  value 
of  w  there  correspond  k  values  of  z,  and  consequently  between  z  and  w 
there  exists  an  integral  algebraic  equation 

G(z,  w)  =  0, 

which  is  of  the  nth  degree  in  w  and  of  the  kih  degree  in  z. 

We   may  next  suppose  that  z  =  </)(u)  is  a  doubly  periodic  function 

with  the  periods  a  and  b,  then  w  =  -^-  =  <t>'(u)  is  a  doubly  periodic  function 

du 

having  the  same  periods.     Hence  from  the  theorem  above  there  is  an 

algebraic  equation  connecting  z  and  —  ,  say 

du 


4!)-°- 


It  is  easy  to  determine  the  degree  of  /in  z  and      ;  for  if  <f)(u)  =  z  is  of 
7  du 

the  nth  degree  then  —  occurs  to  the  nth  degree  in  the  above  equation. 
du 

If  MI  is  an  infinity  of  the  ^th  order  of  (j>(u),  then  MI  is  an  infinity  of  the 
/  -f  1  order  of  <j)'(u),  so  that  <j>'(u)  becomes  infinite  on  the  same  points 
as  <f>(u),  the  order  of  infinity  of  (/>'(u)  being  one  greater  on  each  of  these 
points  than  is  the  order  of  (j>(u)  on  the  same  point. 

If  all  the  infinities  of  </>(u)  are  of  the  first  order  and  if  n  is  the  order  of 
4>(u),  it  follows  that  (/>'(u)  is  of  the  2  nth  order  and  consequently  the 

*  Cf.  Neumann,  Abel'schen  Integrate,  p.  107. 


126  THEORY   OF   ELLIPTIC    FUNCTIONS. 

degree  of  /(  z,  -r\is  at  most  2  n  in  z.      This  equation  f(z,  —  )we  have 
\     du/  \    du) 

called  the  eliminant  equation. 

ART.  106.  In  Art.  104  we  saw  that  any  two  doubly  periodic  functions 
that  have  the  same  periods  .are  connected  by  an  algebraic  equation.  It 
will  therefore  be  sufficient,  if  we  confine  our  attention  to  any  doubly 
periodic  function  and  express  the  others  which  have  the  same  periods 
through  this  one.  This  function  we  shall  take  of  the  second  order  (cf  .  Art. 
92)  and  consequently  either  z  =  sn  u  or  z  =  <@u  (Art.  102). 

Let  z  be  a  doubly  periodic  function  of  the  second  order  (n  =  2),  so  that 
the  eliminant  equation  is  /  dz\ 

•T  Au)  -  °' 

which  is  of  the  second  degree  in  —  and  at  most  of  the  fourth  degree  in  z. 

du 

The  above  equation  must  therefore  have  the  form 


(I)  9o(z)  +  gi(z)     -  +  g2(z)  =  0, 

\duj  du 

where  the  g's  are  integral  functions  of  at  most  the  fourth  degree. 

We  saw  above  that  z  and  —  are  infinite  at  the  same  points  within  the 

du 

period-parallelogram  and  that  -^-  does  not  become  infinite  for  values  of  u 

du 

other  than  those  which  make  z  infinite. 
But  from  (I)  it  is  seen  that 


du  g0(z) 

and  becomes  infinite  for  those  values  of  z  which  make  g$(z)  =  0.  It 
follows  that  QQ (z)  must  be  a  constant  and  consequently  the  equation  (I) 
becomes  /,  N2  j- 

(io      Or)  +  0i<0£f,*(«)-Qp 

\duj  du 

where  the  constant  has  been  absorbed  in  the  two  functions  g\(z)  and  g%(z). 

ART.  107.     If  z  is  a  doubly  periodic  function,  then  also  v  =- is  a  doubly 
periodic  function.     Further,  we  have  at  once 

dz_  _  dz  dv  _        1_  dv . 
du      dv  du          v2  du 

Making  these  substitutions  in  the  above  differential  equation  we  have 
fdv\2  1          fl\dv  1          /IN     n 

fc;^-^vW^P+%ra 

Since    9ri(-jandg2  (-)are  at  most  of  the  fourth  degree  in -,  it  follows 
\vj  W  v 


CONSTRUCTION   OF   DOUBLY   PERIODIC    FUNCTIONS.       127 

that  v4gi(-}  and  v4g2l-  )  are   integral  functions   of   at   most  the   fourth 

W  W 

degree  in  v,  which  we  denote  respectively  by  9i(v)  and  92(0). 

The  above  differential  equation  is  then 


v2    du 

We  saw  above  that  -^  is  finite  for  finite  values  of  z;  the  same  must  also 

fiv  ®  ^ 

be  true  of  —and  v. 


But  in  the  differential  equation  just  written  --  becomes  infinite  for 


v  =  0.     It  follows  that  gi(v)  cannot  be  of  the  fourth  but  must  be  of  the 
second  degree  in  z  at  most. 

It  then  follows  from  the  equation  (F)  that 


=  -  i  9  1  (z)  ±  i  \/gi(z)2-4g2(z)  ; 
du 


or,  if  we  write  4  R(z)  =  g\(z)2  —  4 


du 

where  R(z)  is  an  integral  function  of  at  most  the  fourth  degree. 
It  follows  that  f*z  dz 


f*z  dz 

*      -  *9i(z)  ±  ^R& 


Our  problem  consists  in  the  treatment  of  this  integral  when  R(z)  is  of 
the  third  or  the  fourth  degree;  when  R(z)  is  of  the  second  or  first  degree 
the  integral  is  an  elementary  one. 

/**  rl? 

If  we  write  u  =    I  , 

Jo  V  1  -  z2 

we  have  u  =  sin-1;?,  where  the  inverse  sine-function  is  many-  valued. 

We  know,  however,  that  the  upper  limit  z  considered  as  a  function  of 
the  integral  and  written  z  =  sin  u  is  a  one-valued  simply  periodic  function 
of  u.  In  the  more  general  case  above  we  wish  to  consider  z  as  a  function 
of  u.  This  is  the  so-called  "  problem  of  inversion."  Possibly  the  clearest 
and  simplest  method  of  treating  this  problem  is  in  connection  with  the 
Riemann  surface  upon  which  the  associated  integrals  may  be  represented. 
Before  proceeding  to  the  problem  of  inversion  we  shall  therefore  consider 
this  surface  in  the  next  Chapter. 

EXAMPLE 

1.  If  two  doubly  periodic  functions  /(z)  and  <£(z)  have  only  two  poles  of  the  first 
order  in  the  period-parallelogram  and  if  each  pole  of  the  one  function  coincides 
with  a  pole  of  the  other,  then  is 

t(z)  =  Cf(z)  +  Clf 
where  C  and  C\  are  constants. 


CHAPTER  VI 
THE  RIEMANN  SURFACE 

ARTICLE  108.  At  the  close  of  the  preceding  Chapter  we  were  left  with 
the  discussion  of  an  integral  which  contained  a  radical.  Such  an  expres 
sion  is  two-valued,  and  we  must  now  consider  more  closely  the  meaning  of 
such  functions  and  their  associated  integrals. 

Take  as  simplest  case  the  example 

s  =  ±  v 

where  z  is  a  complex  variable  and  a  an  arbitrary  constant.  For  the  value 
z  =  a,  we  have  s  =  0;  but  for  all  other  finite  values  of  z  there  are  two 
values  of  s  that  are  equal  and  of  opposite  signs.  The  point  a  is  called  a 
branch-point  of  s.  The  point  z  =  oo  is  also  a  branch-point  of  this  function; 

for  -  =  -  =  0  for  z  =  oo .     Consequently  —  and  likewise  s  has 

s       ±  V  z—  a  s 

only  one  value  for  z  =  oo . 

There  are  other  reasons  why  z  =  a  and  z  =  oo  are  called  branch 
points.  Corresponding  to  the  value  z  =  z0,  let  s  =  s6  be  a  definite  value 
of  s.  Along  the  curve  (1)  from  z0  to  zi  consider  the  values  of  s  at  all 
the  points  of  the  curve  which  differ  from  one  another  by  infinitesimally 
small  quantities,  and  similarly  consider  the  values  of  s  along  the  curve 
(2)  until  we  again  come  to  z^  The  value  of  s  at  this  point  will  be  the 
same  whether  we  have  gone  over  the  first  or  second  .curve,  provided  the 
branch-point  a  is  not  situated  between  the  two  curves. 

This  may  be  shown  geometrically  as  follows: 

Let      OM  =  z,  Oa  =  a, 

aM  =  z  —  a,    and    |  z  —  a  \  =  r. 
We  therefore  have 

z  —  a  = 


Fig  27  where  </>  is  the  angle  that  aM  makes  with 

the  real  axis. 
It  follows  that  i$  i$0 

s  =  r^e  2      and      SQ 


If  aM  turns  about  a,  and  M  starting  from  z0  after  making  a  circuit  returns 
again  to  ZQ,  then  if  this  circuit  does  not  include  a,  the  values  of  <f>0  and  SQ 

128 


THE   RIEMANH   SURFACE. 


129 


are  the  same  as  before  the  circuit  and  consequently  s0  has  its  initial  value. 
But  if  the  circuit  includes  a,  the  quantity  r0  is  the  same  after  the  circuit, 
but  <po  has  become  <£0  +  2  -.     The  corresponding 
value  of  so  is 


We  thus  see  that  s0  has  taken  the  opposite  sign 
after  the  circuit. 

ART.  109.     Consider  next  the  expression 
s2  -  R(z), 

where  R(z)  is  an  integral  function  of  the  fourth 
degree  in  z.     We  may  write 


Fig.  28. 


R(z)  =  A(z-  Ol)  (z  -  a2)  (z  -  a3)  (z  -  a4), 

A  being  a  constant. 
We  then  have 

s  =  ±  VR(z)  =  ±  A*  (z  -  d)*  (z  -  a2)4  (z  -  a3)*  (z  -  o4)*- 

The  function  s  has  two  values  with  opposite  signs  for  any  value  of  z  except 
°i>  a2,  ^3,  «4-     When  z  is  equal  to  any  of  these  values,  s  has  the  one  value 
zero.     The  points  a  i,  a2,  «s  and  a4  are  branch-points.     The  value    a±  —  ZQ 
is  the  radius  of  the  circle  about  ZQ  wrhich  goes  through  a\.     Suppose  that 
z  is  any  point  situated  within  this  circle  so  that 

\Z  -  ZQ\    <    \di   -  ZQ\. 

Then,  since  z  —  ai  =  z  —  ZQ  —  (ax  —  z0),  we  have 


(z  - 


=        -  (a, 


-  z       1  -  ^^^     *. 


Since 


<  1,  the  right-hand  side  may  be  expanded  by  the  Binomial 


Theorem  in  the  form 


-  ZQ 


di   -  ZQ 


This 'series  is  uniformly  convergent  for  all  values  of  z  within*  the  circle. 
In  the  same  wTay  we  may  develop  (z  —  a2)*,  (z  —  ^3)*,  (z  —  a4)*  in  posi 
tive  integral  powers  of  z  —  ZQ.  All  these  series  are  convergent  within 
circles  about  ZQ. 

We  have  the  development  of  s  in  powers  of  z  —  ZQ  by  multiplying  the 

*  When  we  say  "within"  we  mean  within  any  interval  that  lies  wholly  within. 
See  Osgood,  loc.  tit.,  p.  77  and  p.  285. 


130 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


four  series  together,  the  multiplication  being  possible,  since  the  series  of 
the  moduli  of  the  terms  that  constitute  the  four  series  are  convergent. 
We  thus  derive  the  result:  We  may  develop  s  =  \/R(z)  in  positive  integral 
powers  of  z  —  ZQ,  if  z0  is  different  from  the  four  branch-points  ai}  a2,  a3,  a4. 
The  series  is  uniformly  convergent  within  the  circle  about  ZQ  as  center,  which 
passes  through  the  nearest  of  the  points  «1?  a2,  a3,  a4. 

ART.  110.     We  may  effect  within  this  circle  the  same  development  by 
Taylor's  Theorem  in  the  form 


2       S0 

We  must  decide  upon  a  definite  sign  of  s0  =  VR(z^  and  use  this  sign 
throughout  the  development.  If  at  the  beginning  we  decide  upon  the 
other  sign,  then  in  the  series  we  must  write  —  s0  instead  of  s0;  that  is, 
all  the  coefficients  are  given  the  opposite  sign. 

If  the  sign  of  s0  has  been  chosen  and  if  the  development  of  s  has  been 
made,  then  s  is  defined  through  the  above  series  only  within  the  circle 
already  fixed.  If  we  consider  a  value  of  z  without  the  circle  of  convergence, 
we  do  not  know  what  value  s  will  take  at  this  point.  To  be  more  explicit 
we  may  proceed  as  follows: 

Let  z'  be  a  point  without  the  circle  and  join  z*  with  ZQ  through  any 
path  of  finite  length  which  must  not  pass  indefinitely  near  a  branch 
point.  Let  the  circle  of  convergence  about 
ZQ  cut  this  path  at  £.  Then  at  all  points  of 
the  portion  of  path  20£  the  corresponding 
values  of  the  function  are  known  through 
the  series.  Let  z\  be  a  point  on  this  portion 
of  path  which  lies  sufficiently  near  to  the 
periphery  of  the  circle.  We  may  express  the 
value  of  the  function  at  zif  that  is,  si  = 
V  R(ZI)  through  the  series 

si  =  ' 


|2=2l 


i    29. 


Thus  si  is  uniquely  determined,  if  the  sign  of 
s0  has  been  previously  chosen. 
We  next  take  z\  as  the  center  of  another  circle  Ci,  which  also  must  not 
contain  a  branch-point.  Then  precisely  as  we  expanded  5  in  powers  of 
z  —  ZQ  in  the  circle  Co  about  ZQ  we  may  now  expand  s  within  C\  in  powers 
of  z  —  z\  about  z\.  This  circle  C\  may  extend  up  to  the  nearest  branch 
point  and  is  not  of  an  infinitesimally  small  area,  since  by  hypothesis 
the  path  did  not  come  indefinitely  near  a  branch-point.  The  point  z\ 
is  taken  sufficiently  near  £  that  the  circle  about  z\  partly  overlaps  the 


THE   BIEMANN    SURFACE.  131 

circle  about  ZQ.  That  this  may  be  the  case  z\  must  lie  so  close  to  £  that 
the  distance  between  the  points  is  less  than  the  radius  of  the  circle  C\, 
a  condition  which  evidently  may  always  be  satisfied.  Hence  the  circles 
Co  and  Ci  have  a  portion  of  area  in  common.  Let  the  power  series  which 
is  convergent  within  Co  be  denoted  by  PQ(Z  —  ZQ)  while  the  one  in  C\ 
may  be  represented  by  PI  (z  —  z\).  As  we  have  already  seen  in  Chapter  I 
the  series  PI  gives  for  every  value  of  z  which  is  common  to  the  two  circles 
the  same  value  as  does  the  series  P0.  But  the  development  PI  holds 
good  for  the  entire  circle  Ci.  We  thus  go  in  a  continuous .  manner  to 
values  of  the  function  which  lie  without  the  circle  Co-  The  series  PI 
represents  the  continuation  of  the  function  s. 

It  is  clear  that  this  process  may  be  repeated  and  that  we  will  finally 
come  to  a  circle  Cm  around  a  point  zm  of  the  path  as  center  within  which 
the  point  z'  lies.  We  may  develop  the  function  within  Cm  in  positive 
integral  powers  of  z  —  zm  and  may  then  compute  sf  =  \/R(z*)  from  this 
development.  This  process  is  called  the  "  Continuation  of  the  function 
along  a  prescribed  path  from  ZQ  to  z'."  Such  a  continuation  is  possible  in 
the  entire  z-plane,  since  ZQ  may  be  connected  by  such  a  path  with  any 
other  point  z  which  is  not  a  branch-point. 

ART.  111.  Let  B  and  BI  be  two  different  paths  which  join  ZQ  and  z' 
and  suppose  that  neither  of  these  points  lies  indefinitely  near  a  branch 
point.  The  question  arises  whether  the  value  of  the  function  at  zf  which 
is  had  through  the  continuation  of  the  function  along  the  path  B\  is  the 
same  as  the  one  which  is  had  through  the  continuation  from  ZQ  to  /  alon^ 
B.  It  is  clear  that  if  the  two  values  of  \/R(zf)  thus  ob 
tained  are  different,  they  can  differ  only  in  sign. 

Through  the  circles  which  are  necessary  for  the  con 
tinuation  of  the  function  from  ZQ  to  zf  along  B  is  formed 
a  strip  (see  figure  of  preceding  article)  which  has  every 
where  a  finite  breadth.  This  strip  may  be  regarded  as 
a  "  one-value  realm."  The  function  s  remains  one-valued 
within  this  realm.  First  suppose  that  the  path  BI  lies 
also  wholly  within  this  realm. 

Since  none  of  the  circles  contains  a  branch-point 
there  cannot  be  one  between  B  and  BI,  and  it  is  evident  T-  on 

that  we  come  through  the  continuation  of  the  function 
along  these  curves  to  the  point  z'  with  the  same  value  of  the  function. 
For  let  the  normal  at  any  point  ak  on  B  cut  the  curve  BI  at  «&'  where 
BI  is  taken  very  near  to  B.  as  shown  in  Fig.  31,  and  call  ak,  ak  a  pair 
of  neighboring  points. 

We  suppose  that  the  curves  B  and  B  i  have  been  taken  so  near  together 
that  one  of  the  circles  employed  in  the  continuation  of  the  function  along 
B  contains  both  ak  and  ak  and  that  all  points  within  this  circle  are  ex- 


132 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


Fig.  31. 


pressed  through  Pk(z  —  Zk)'}  and  at  the  same  time  we  assume  that  one  of 
the  circles  used  in  the  continuation  of  the  function  along  the  path  BI 
includes  also  the  same  points  ctk,  <x.il  and  that  all  points  within  this  circle 
are  had  through  the  series  Pv(z  —  zv).  Hence  we  must  have  the  same 

value  of  s  at  the  point  z  =  ak  from  either 
of  the  power  series  P^  or  Pkf,  provided  this 
is  true  of  every  pair  of  neighboring  points 
that  preceded  this  pair.  The  same  is  also 
true  of  the  point  z  =  a^.  But  the  first 
pair  of  neighboring  points  was  the  point 
ZQ.  We  therefore  come  to  z'  with  the 
same  value  of  s  along  either  path  B  or  BI. 
Heffter  [Theorie  der  Linear  en  Differential- 

Gleichungen,  p.  72]  has  given  a  somewhat  similar  proof  which  suggested  the 
one  given  here  [see  my  Calculus  of  Variations,  pp.  15,  16  and  256  et  seq.]. 
If  next  B  and  BI  are  two  curves  which  are  drawn  in  an  arbitrary  manner 
between  ZQ  and  z' ,  but  which  do  not  include  a  branch-point,  then  we  may 
fill  the  surface  between  B  and  B\  with  a  finite  number  of  curves  drawn  from 
ZQ  to  zf  which  lie  at  a  finite  distance  from  one  another 
and  are  so  situated  that  each  one  lies  within  the 
one-valued  realm  which  is  formed  by  the  circles 
that  are  necessary  for  the  continuation  of  the  func 
tion  along  a  neighboring  curve.  Thus  by  means  of 
the  intermediary  curves  with  their  associated  one- 
valued  realms  it  is  evident  that  we  come  to  z'  with 
the  same  value  of  s  when  we  make  the  continuation 
along  either  of  the  two  curves  B  or  BI  provided 
that  there  is  no  branch-point  between  them.  It 
follows  also  that  the  value  of  the  function  at  the 
point  z'  is  independent  of  the  form  of  the  curve 
between  ZQ  and  zf. 

ART.  112.     Let  (1)  and  (2)  be  two  paths  between  ZQ  and  z'  which  do  not 
include  a  branch-point.     If  we  go  along  (2)  from  00  to  z'  and 
then  back  again  along  (1)  from  z'  to  ZQ,  we  come  to  the  same 
initial  value   of  the  function,     From  this  it  follows:  //  the 
^  function  s  =  \/R  (z)  is  continued  from  the  point  z  =  ZQ  along  a 
I     closed  curve  which  does  not  contain  a  branch-point,  we  return 
after  the  circuit  to  the  point  ZQ  with  the  same  initial  value  of 
the  function. 

The  form  of  the  curve  is  arbitrary,  provided  only  it  does 
not  inclose  any  branch-point.  Hence  instead  of  making  a 
circuit  around  an  arbitrary  curve,  we  may  choose  a  circle  which  passes 
through  ZQ. 


Fig.  32. 


Fig.  33. 


THE    RIEMAXX    SURFACE.  133 

ART.  113.     Suppose  next  that  the  closed  curve  includes  a  branch-point, 
for  example  ai.     We  again  fix  the  sign  of  s0  for  z  =  zQj  and  write 


s  =  Vz  -  alVRl(z), 
where 

A(z  -  a2)  (z  -  a3)  (z  -  a4). 


We  may  allow  VRi(z)  to  have  an  arbitrary  sign,  and  so  choose  the  sign 
of  \/z  —  ai  that  s  =  s0  will  have  the  same  sign  for  z  =  z0  as  has  been  pre 
viously  assigned  to  it.  _ 

If  we  make  a  circuit  about  a  i,  it  is  seen  that  \/Ri(z)  is  not  affected  by 
it,  since  ai  is  not  a  branch-point  of  \/R\(z).  Hence  upon  making  a 
circuit  about  01  we  need  consider  only  the  first  factor  (z  —  ai)*.  We 
may  make  this  circuit  along  a  circle  of  radius 
r  with  a  i  as  center.  For  the  points  of  the 
periphery,  it  is  clear  that 

|  z  -  a1  |  =  r, 

so  that 

z  —  a  i  =  re^. 

It  follows  that  ^ 

(Z  -  Ol)i  =  r*e*. 

Let  the  value  of  <j>  corresponding  to  z  =  z0  be 
<£  =  (f>Q,  so  that  t£  Fig.  34. 

(z0  -  ai)*  =  r*e2, 

where  the  point  ZQ  of  course  lies  upon  the  periphery  of  the  circle.  When 
a  complete  circuit  is  made  about  01,  starting  from  z0,  it  is  seen  that  <£o 
is  increased  by  2  -,  and  consequently  after  this  circuit  the  above  expres 
sion  becomes 


rie      2        _  r*e  2  ci*  =  -  r*e  2  . 
It  follows  that  after  a  circuit*  about  QI  has  been  made,  the  quantity 


(z  —  ai)*  and  consequently  also  s  =  VR(z)  changes  its  sign. 

Further,  if  we  make  a  circuit  about  a!  along  any  arbitrary  curve  B 
which  does  not  include  any  other  branch-point  except  «i,  then  s  changes 
sign  with  this  circuit;  for  this  is  the  case  when  a  circuit  has  been  made 
about  the  circle  around  01,  and  as  there  is  no  branch-point  between  the 
circle  and  the  path  B,  it  follows  that  starting  from  z0  ^"e  will  again  return 
to  this  point  along  both  of  the  curves  with  the  same  value  of  the  function. 

ART.  114.  We  may  next  ask  what  happens  if  the  circuit  includes  two 
branch-points.  First  suppose  that  the  circuit  is  made  along  the  path 
z0a3fZQ.  Let  os-  be  a  closed  curve  about  ai  and  yOn  a  closed  curve 
about  a2.  It  follows  immediately  from  the  above  considerations  that  the 

*-Cf.  Bobek,  Elliptische  Functionen,  p.  150. 


\\ 


134  THEORY   OF   ELLIPTIC    FUNCTIONS. 

two  curves  between  which  there  is  no  branch-point  lead  always  to  the 
same  initial  value  of  the  function. 

Hence  instead  of  making  the  circuit  about  a\  and  a2  along  the  path 
o  we  may  just  as  well  make  the  circuit  along  the  path  z0d 'srzoyd KZQ, 
there  being  no  branch-point  between  this  curve 
and  the  curve  z0a^z0.  After  the  circuit 
z0d£TZo  the  function  s  changes  sign  as  it  again 
does  after  the  circuit  ZQ^KZQ,  so  that  after  the 
two  circuits  around  the  points  a{  and  a2  we 
again  come  to  the  point  ZQ  with  the  initial  value 
of  s. 

We  conclude  in  the  same  way  that  if  we 
make  an  arbitrary  circuit  around  four  branch 
points  we  again  come  to  the  same  value  of  the 
function,  while  if  we  have  encircled  three  branch 
points,  we  arrive  at  z0  with  the  other  value  of  s. 
ART.  115.  We  may  next  see  how  the  function 


s  =  A*  \/{z  —  a\)(z  —  o2)  .  .  .   (z  —  an  ) 

behaves  when  a  circuit  is  made  around  the  point  at  infinity.  When  n 
is  an  even  integer  and  when  a  circuit  is  made  so  as  to  include  the  n  points 
01,  02,  .  .  .  ,  an,  it  follows  from  above  that  when  z  returns  to  its  initial 
position,  the  value  of  s  has  not  changed  its  sign.  In  the  above  expression 

write  z  =  — ,  so  that  when  z  =  GO  ,  we  have  t  =  0.     In  the  z-plane  the  point 
t 

at  infinity  corresponds  to  the  origin  in  the  £-plane.     We  then  have 


s  =  t    2  A  K/(l  -  aiQ  (1  -  a2t)  ...   (1  -  ant). 


Now  take  a  circuit  about  a  circle  with  the  origin  as  center  and  which 
does  not  contain  one  of  the  branch-points  01,  02,  -  •  •  ,  on.  We  must 
therefore  write 

*  == 


and  it  is  seen  that  the  function  s  changes  sign  when  n  is  an  odd  integer. 
In  this  case  the  origin  in  the  Z-plane  is  a  branch-point,  and  consequently 
in  the  z-plane  the  point  at  infinity  is  or  is  not  a  branch-point  according  as 
n  is  an  odd  or  even  integer. 

ART.  116.  We  shall  draw  lines  connecting  the  points  a\  with  a2  and  a3 
with  04.  The  paths  along  which  the  function  s  is  continued  must  never 
cross  these  lines  a\  a2  and  03  04.  They  may  be  called  "  canals."  The 
z-plane  which  contains  these  two  canals  may  be  denoted  by  the  z-plane, 
a  dash  being  put  over  z  (see  Fig.  36). 


THE   KIEMAXX    SURFACE. 


135 


If  once  the  initial  value  s0  of  the  function  s  =  \/R(z)  is  fixed  for  the 
point  ZQ,  then  s  is  completely  one-valued  in  the  z-plane;  for  in  whatever 
manner  the  continuation  from  z0  to  zf 
may  be  made,  any  two  different  paths 
will  always  include  an  even  number  of 
branch-points  or  none,  since  the  canals 
cannot  be  crossed.  It  follows  that 
s  =  \/R(z)  no  longer  depends  upon  the 
path  along  which  this  function  is  con 
tinued  from  one  point  to  another  and 
is  consequently  one- valued  in  the  z-plane. 
The  two  canals  are  sometimes  called 
branch-cut*. 

If  further  the  sign  has  been  ascribed  to  the  initial  value  s0  of  the 
function  s,  then  we  may  ascribe  to  s  its  proper  value  for  every  value  in  the 
2-plane.  These  values  we  suppose  have  been  written  down  on  a  leaf, 
which  represents  the  z-plane.  Again  starting  with  -  s0  for  the  initial 
point  we  consider  the  corresponding  values  of  the  function  written  down 
upon  another  plane  or  leaf.  In  this  second  leaf  the  two  canals  connecting 
a i  with  e&2  and  a3  writh  a4  are  also  supposed  to  have  been  drawn,  so  that  s 
is  also  one-valued  on  it. 

We  note  that  corresponding  to  the  same  value  of  z,  the  values  of  s  = 
±vR(z)  in  the  two  leaves  are  equal  but  of  opposite  sign.  If,  further, 
starting  from  a  point  ai  on  the  upper  bank  of  the  canal  we  make  a  circuit 


Fig.  37. 


around  a\,  say,  and  return  to  the  point  «2  immediately  opposite  on  the 
lower  bank,  the  values  of  s  at  these  two  points  are  the  same  with  con 
trary  sign.  The  same  is  true  for  all  points*  opposite  one  another  along  the 
two  canals  a^  a2  and  a3  a4. 

We  imagine  the  two  leaves  placed  the  one  directly  over  the  other, 
with  the  canals  in  the  one  leaf  over  those  in  the  second  leaf.     The  left 


Cf.  Neumann,  Abel'schen  Integrate,  p.  81. 


136 


THEOKY    OF   ELLIPTIC   FUNCTIONS. 


bank  of  each  canal  in  the  upper  leaf  is  joined  with  the  right  bank  in  the 
lower  leaf  and  the  right  bank  in  the  upper  leaf  with  the  left  bank  in  the 
lower.  If  being  in  the  upper  leaf  we  cross  a  canal  we  will  find  ourselves 
in  the  lower  leaf;  and  if  being  in  the  lower  leaf  we  cross  a  canal  we  will 
come  up  in  the  upper  leaf.  Thus  the  values  of  the  function  s  change  in  a 
continuous  manner  when  by  crossing  the  canals  we  go  from  one  leaf  into 
the  other;  and  in  this  manner  we  are  able  to  make  the  two-valued  function 
s  behave  like  a  one-valued  function  by  means  of  the  above  structure.  In 
this  structure  there  is  no  crossing  from  one  leaf  to  the  other  except  in  the 
manner  indicated  by  means  of  the  canals. 

The  structure  is  called  the  Riemann  surface  *  of  the  function  s  = 


(cf.  Grundlagen  fur  eine  allgemeine  Theorie  der  Funktionen  einer  kom- 
plexen  verdnderlichen  Grosse.  Inauguraldissertation  von  B.  Riemann. 
Crelle,  Bd.  54,  pp.  101  et  seq.). 

If  the  function  is  continued  anywhere  in  this  Riemann  surface,  the 
function  has  always  at  any  definite  point  a  definite  value,  which  is  indepen 
dent  of  the  path  along  which  the  function  has  been  continued.  It  is  thus 
shown  that  the  function  s  is  a  one-valued  function  of  position  in  the  Riemann 
surface.  In  this  surface,  if  for  a  definite  value  of  z  the  corresponding  value 
of  s  is  to  be  found,  we  must  also  indicate  whether  the  value  of  z  is  taken 
in  the  upper  or  in  the  lower  leaf. 


7  a2 


Fig.  38. 

In  the  figures  a  path  that  is  taken  in  the  lower  leaf  is  denoted  by  a  broken 
line  ( : ),  while  a  path  in  the  upper  leaf  is  indicated  by  an  uninter- 

*  See  also  Neumann,  Theorie  der  Abel'schen  Integrate;  Durege,  Elemente  der  Theorie 
der  Funktionen.  For  other  references  see  Wirtinger,  Ency.  der  math.  Wiss.,  Bd.  II3, 
Heft  1, 


THE   KIEMANX    SUEFACE.  137 

rupted  line  (  _  ).  The  fact  that  the  function  s,  when  a  circuit 
is  taken  around  no  branch-point,  or  around  two  branch-points,  or  around 
four  branch-points,  retains  its  sign,  while  it  changes  sign  if  the  path  is 
around  one  or  three  such  points,  is  brought  into  evidence  by  means  of  the 
Riemann  surface.  It  is  indicated  in  the  figures  on  page  136. 

We  no;te  that  by  a  circuit  around  one  or  three  branch-points  we  always 
pass  from  one  leaf  into  the  other,  and  that  at  two  points  situated  the  one 
over  the  other  the  function  s  has  the  same  absolute  value  but  different 
signs. 

THE  ONE-  VALUED  FUNCTIONS  OF  POSITION  ON  THE  RIEMANN  SURFACE. 

ART.  117.  We  have  denned  a  function  as  being  one-  valued  on  the 
Riemann  surface.  We  may  now  consider  more  closely  what  is  meant 
by  such  a  function.  When  we  say  that  a  function  is  "  one-  valued  on  the 
Riemann  surface/'  we  mean  something  quite  different  from  what  is  meant 
by  saying  a  "  function  is  one-valued."  The  signification  of  the  first  defini 
tion  is:  "If  the  value  of  the  variable  z  is  given  and  also  the  position  on 
the  Riemann  surface,  then  the  function  is  uniquely  determined";  if,  however, 
only  z  were  given,  the  function  would  not  be  uniquely  determined. 

Let  w  be  any  function  whatever  of  z  which  we  suppose  is  one-valued 
on  our  fixed  Riemann  surface.  In  the  upper  leaf  of  this  surface  the  function 
w  has  for  a  given  z  a  definite  value,  say  wi,  and  in  the  lower  leaf  it  takes 
another  value,  say  w2j  for  the  same  value  of  z.  In  the  special  case  above 
where  w  =  s  =  ±  vR(e)t  we  have  w\  =  —  w2.  In  general,  however, 
this  is  not  the  case.  But  if  we  consider  the  sum  wi  +  w%,  this  sum  is  a 
one-valued  function  of  2,  for  if  z  is  given,  w\  +  w2  is  completely  determined. 
The  same  is  also  true  of  the  product  wi  •  w2. 

It  follows  that  w  satisfies  a  quadratic  equation  of  the  form 

w2  -  <}>(z)w  +  ^0)  =  0, 
where  (j>(z)  and  ty(z)  are  one-valued  functions  of  z,  such  that 

wi  +  w2  =  <j)(z)     and     wi  •  w2  = 


Hence  every  one-valued  function  of  position  on  the  Riemann  surface 
s  =  vR(z)  is  a  two-valued  function  of  z  and  satisfies  a  quadratic  equation, 
whose  coefficients  are  one-valued  functions  of  z. 

In  particular,  we  shall  study  those  one-valued  functions  of  position  on 
the  Riemann  surface  which  have  a  definite  value  at  every  position  on  the 
Riemann  surface.  In  this  case  <j)(z)  =  w\  +  w2  will  have  a  definite  value 
for  every  value  of  z,  as  will  also  ^r(z)  =  w\  •  w2.  But  one-  valued  functions 
which  have  everywhere  definite  values  (when  therefore  there  is  no  essen 
tial  singularity)  are  rational  functions,  If  then  w  is  to  be  a  one-valued 


138  THEORY    OF   ELLIPTIC    FUNCTIONS. 

function  of  position  on  the  fixed  Riemann  surface  and  is  to  have  every 
where  on  this  surface  a  definite  value,  then  <p(z)  and  ^r(z)  must  be  rational 
functions  of  z. 

ART.  118.     When  we  solve  the  above  quadratic  equation,  we  have 


where  the  root  is  to  be  taken  positive  or  negative.  We  have  thus  shown 
that  w  is  equal  to  a  rational  function  of  z,  increased  or  diminished  by  the 
square  root  of  a  rational  function. 

Suppose  that  the  radicand  —  4  -^(2)  +  c/)2(z)=  S(z),  say,  becomes  zero 
or  infinite  of  the  (2  n  +  l)st  order  for  z  =  b,  where  n  is  an  integer. 

We  note  that  the  point  6  cannot  be  a  branch-point  on  the  Riemann 
surface,  for  ai,  a2,  a^,  #4  are  the  only  branch-points  on  this  surface. 

We  may  write  S(g)  =  (z  _  b)2  m+ig^ 

where  Si(z)  is  a  rational  function  of  z. 

About  b  as  a  center  describe  a  circle  which  does  not  inclose  any  other 
zero  or  infinity  of  S(z). 

We  then  have  2n+i 


and  if  z  makes  a  circuit  about  the  circle,  the  function  V$  ±(z)  retains  its 

2n  +  l 


sign,  while  (z  —  b)    2      changes  sign.     Consequently  the  function 


changes  its  sign  with   this   circuit,  so   that   w  =  2SS  -| does  not 

resume  its  initial  value  and  is  therefore  not  a  one-valued  function  of 
position  on  the  Riemann  surface.  Hence  the  factor  z  —  b  must  occur 
to  an  even  power  if  it  enters  as  a  factor  of  either  the  numerator  or  the 
denominator  of  the  rational  function  S(z),  so  that  S(z)  must  have  the 
form 

S(z)  =  Si(z)2  { (z  -  ai)  (z  -  a2)  (z  -  a3)  (z  -  o4) }- 
We  may  therefore  write 


w 


=  $  <f>(z)  +  %  Si(z)  V(z  -  ai)  (z  -  a2)  (z  -  a3)  (z  -  o4) 
=  p(z)  +  q(z)  VR(z)  =  p  +  q  •  s, 


where  p  =  p(z)  =          ,    q  =  q(z)=  k  l^  are  rational  functions  of  z. 

£  2/ 

It  has  thus  been  shown  *  that  "  Every  one-valued  function  of  position, 
which  has  everywhere  a  definite  value  in  our  Riemann  surface,  is  of  the  form 

w  =  p  +  qs, 
where  p  and  q  are  rational  functions  of  z." 

*  Cf.  Neumann,  loc.  cit.,  p.  355. 


THE   RIEMANN    SURFACE.  139 

Reciprocally,  every  function  of  the  form  w  =  p  +  qs  is  a  one-valued 
function  of  position  on  the  Riemann  surface,  since  p,  q,  s  taken  separately 
have  this  property.  If  then  w  has  this  form,  it  is  the  necessary  and 
sufficient  condition  that  it?  be  a  one-valued  function  of  position  on  the 
Riemann  surface. 

THE  ZEROS  OF  THE  ONE-VALUED  FUNCTIONS  OF  POSITION. 

ART.  119.  Let  z  =  a  be  a  position  on  the  Riemann  surface,  which  is 
different  from  the  branch-points  «i,  a2,  a3,  a4.  We  may  then  draw  a 
circle  around  a  which  lies  entire!}7  in  one  leaf  of  the  Riemann  surface. 

It  may  happen  that  w  =  0  for  z  =  a,  while  at  the  same  time  p  and 
q  are  infinite  for  z  =  a.  For  suppose  that 


(z  —  a)*4"1  z  —  a 

We  may  also  develop  s  for  points  within  the  circle  in  the  form 
s  =  h0  +  hi(z  -  a)  +  h2(z  -  a)2  +  •   • 

It  is  evident  that  s  is  not  infinite  for  z  =  a,  and  it  is  also  clear  that  if 
^  T^  fi,  then  w  becomes  infinite  for  z  =  a;  but  if  ^  =  ,«,  then  we  may  so 
choose  the  coefficients  in  the  development  of  p  and  q  that  w  =  0  for 
z  =  a.  This  will  be  the  case  if  in  the  development  of  w  all  the  negative 
powers  and  also  the  constant  term  drop  out.  The  coefficient  of  (z  —  a)  ~  * 
in  this  development  is  e^  +  h0fu,  or,  since  X  =  (u,  we  must  have 


e,  +  h0f,  =  0. 
Further,  it  is  necessary  that  the  coefficient  of  (z  —  a)-^"1)  be  zero,  that  is, 

ex-i  +f*-ih0  +fihl  =0,     etc. 
These  conditions  may  all  be  satisfied;  and  consequently 

w  =  kr(z  -  a)r  +  kr+i(z  -  a)r+1  +  •  •  •  , 

where  the  k's  are  constant  and  where  r  is  a  positive  integer  greater  than  0. 
Finally  we  may  write 

w  =  (z  -  a)r[kr  +  kr+1(z  -«)  +  •••]. 

We  see  that  w  becomes  zero  of  the  rth  order  for  z  =  a.  We  thus  experi 
ence  no  trouble  in  determining  the  order  of  zero  for  w  at  any  point  a,  even 
if  at  this  point  the  functions  p  and  q  become  infinite.  Similarly  if  p  and  q 
remain  finite  for  z  =  a  there  is  no  difficulty. 


140  THEORY    OF   ELLIPTIC    FUNCTIONS. 

ART.  120.  We  shall  next  study  w  in  the  neighborhood  of  one  of  the 
branch-points,  a\  say.  If  z  makes  a  circuit  about  a\,  we  return  with  a 
value  of  w  that  lies  in  the  other  leaf,  and  in  order  to  reach  the  initial  point 
of  the  circuit  we  must  make  a  double  circle  about  ai,  since  by  the  second 

circuit  we  again  come  into  the  leaf  in  which  the 
initial  point  is  situated. 
As  in  Art.  113,  we  write 

s  =  VR&  =  (z  - 


Since  a\  is  not  a  branch-point  of  y/Ri(z)t  we 
may  expand  this  function  in  positive  integral 
powers  of  z  —  a\  and  have 

=  (z  —  ai)*[&o  +  bi(z  —  ai)  +  b2(z  —  ai)2  +•••]. 
We  put 

(z  —  ai)*=  t     or     t2  =  z  —  ai. 

Let  a  circuit  be  made  about  a\  along  a  circle  with  radius  r,  so  that 

z  -  al  =  t2  =  re**, 
or  i(j> 

t  =  Vre~*. 

If  then  z  describes  a  circle  with  radius  r  around  ai  in  the  2-plane,  then 
t  describes  a  circle  with  radius  Vr  around  the  origin  in  the  2-plane.  If 
the  circuit  of  z  begins  with  the  initial  value  0  =  0,  then  the  circuit  of  t 
begins  with  the  value  0  =  0,  and  when  0  increases  by  2  n  we  have  0/2 
increased  by  TT.  Hence  to  the  whole  circle  in  the  2-plane  there  corresponds 
the  half-circle  in  the  2-plane,  and  to  the  double  circle  which  z  describes  in 
the  Riemann  surface  in  order  to  return  again  to  its  initial  point,  there 
corresponds  the  simple  circle  in  the  2-plane. 

Suppose  that  w  vanishes  at  a  branch-point,  a\  say.  Further  suppose  by 
the  substitution  z  —  a\  =  t2thatp(z)  becomes  p(t)  and  q(z)  becomes  q(t). 

In  the  neighborhood  of  the  point  t  =  0,  let 

p(t)  =  antm  - 
and  q(t)  =  f3ntn  -f 

where  m  and  n  are  integers  (positive  or  negative  including  zero). 

If  m  and  n  take  negative  or  zero  values,  there  must  exist  equations  of 
condition  as  in  the  preceding  article. 

Since  z  —  a\  =  t2,  it  follows  that  z  —  a2  =  ai  —  a2  +  t2,  z  —  a3  =  ai  — 
«3  +  t2,  z  —  0,4  =  «i  —  04  +  t2,  and  consequently  R\(z)  becomes  V(t), 
where  V(t)  =  fa  -  a2  +  t2)  fa  -  a3  +  t2)  (ai  -  a±  +  t2). 

We  note  that  this  function  does  not  vanish  for  t  =  0,  so  that  there  is  no 
branch-point  of  this  function  within  the  circle  t  =  0,  if  this  circle  is  taken 


THE   RIEMAXN    SURFACE.  141 

sufficiently   small.     We    may    consequently   expand    VV(t)  within    this 
circle  in  positive  integral  powers  of  t2  and  have 

=  t[b0  +  M2  +  M4  +•••]• 


It  is  further  seen  that  if  w  becomes  zero  at  the  point  z  —  0,1  =  ft,  it  may 
be  developed  in  the  neighborhood  of  t  =  0  in  positive  integral  powers  of 
t  in  the  form 


w  =  Co         ci 

A  A+l  A+2 

=  cQ(z  -  di)2+  ci(z  -  aO   2    +  c2(z-al)   2    +....' 

It  follows  also  that  the  function  w  becomes  zero  of  the  ;Uh  order  at  the 
branch-point  z  =  01.  In  other  words,  if  w  becomes  zero  at  a  branch-point 
z  =  ak,  then  TWICE  the  exponent  of  the  lowest  power  of  z  —  ak  in  the  develop 
ment  of  w  in  ascending  powers  of  this  quantity,  is  the  ORDER  of  the  zero  on 
this  position.  If,  however,  the  zero-position  z  =  a,  say,  is  NOT  a  branch 
point,  we  have  the  development 


and  here  the  exponent  of  the  LOWEST  power  of  z  —  a  in  the  development  in 
ascending  powers  'of  this  quantity  is  the  order  of  the  zero  of  the  function  at 
z  =  a. 

This  difference  respecting  the  order  of  the  zeros  seems  at  first  arbitrary, 
but  the  significance  is  evidenced  through  the  following  consideration: 
Let  a  be  a  zero  which  does  not  coincide  with  one  of  the  branch-points. 
We  may  then  develop  w  in  the  form 

w=  (z-  a)*'  [c0'  +  ci'(2  -  a)  +  c2'(z  -  a)2  +••-], 
and  consequently 

log  w  =  x'log(*  -  a)  +  log  [c0'  +  Ci'(z  -  a)  +  c2'(z  -  a)2  +•••]. 

Since  the  expansion  within  the  bracket  does  not  become  zero  for  z  =  a, 
its  logarithm  is  not  negative  infinity  and  the  expression  may  be  developed 
in  integral  powers  of  z  —  a  .  We  then  have 

log  w  =  /'  log  (z  —  a)  +  e\    +  e2(z  —  a)  -f  •  •   •    . 

If  z  makes  a  complete  circuit  about  a,  the  power  series  e\  +  e2'  (z  —  a)  +  .  .  . 
does  not  change  sign;  log  (z  —  a)  is,  however,  increased  by  2  id  and  con 
sequently  /'  log  (z  —  a)  is  increased  by  2  idX. 
It  follows  that  -. 


is  increased  by   X'  when   a   circuit  is   made   about   the   zero   z  =  a:   in 
other  words,  the  order  of  the  zero  of  the  function  w  at  the  point  z  =  a  is  the 


142 


THEOKY   OF   ELLIPTIC    FUNCTIONS. 


number  due  to  the  change  in  -  r  log  w  when  z  makes  an  entire  circuit 

Jj  711 

about  a. 

This  same  analytic  property  must  be  retained  if  a  is  also  a  branch 
point,  say  a  i. 

From  the  development  above 


w  =  (z  —  ai)2[c0  +  ci(z  —  ai)*  +  c2(z  — 
it  follows  that 


log 


log  [c0 


or 


/I 


log  w  =  -log  (z  — 
2 


+  eo  +  e\  (z  — 


experienced  in  -  .  log  w  is  X  since  log  (z  — 
2  TCI 


Now  to  make  a  complete  circuit  around  &i  we  must  make  a  double  circle. 
By  this  circuit  (z  —  01)*  does  not  change  sign.     It  follows  that  the  change 

changes  by  2  •  2  ni.     But 

here  X  is  twice  the  exponent  of  the  lowest  power  of  z  —  a\  in  the  above 
expansion  of  w. 

The  infinities  of  w  may  be  treated  in  precisely  the  same  way  as  its 
zeros. 

INTEGRATION. 

ART.  121.  We  shall  next  consider  the  integrals  taken  over  certain 
paths  in  the  Riemann  surface.  These  are  formed  in  the  same  manner  as 
are  the  integrals  of  functions  of  the  complex  variable  in  the  plane. 

If  w  =  f(z)  =  p  +  qs  is  a  function  which  for  all  points  of  the  path  of 

z\s'  integration  takes  finite  and  con 
tinuous  values,  and  if  a  definite 
path  of  integration  is  prescribed 
which  is  taken  from  the  point 
ZQ,  where  \/R(z)  takes  the  value 
s0,  to  the  point  2',  where 


Fig.  40. 


takes  the  value  s',  then  the  inte 
gral  lf(z)dz  taken  over  this  path 

has  a  definite  value.  If  a  portion 
of  the  path  of  integration  lies  in  the  lower  leaf,  the  significance  is  that  the 
function  under  the  integral  sign  takes  values  in  the  lower  leaf  which  form  a 
continuous  connection  with  the  values  in  the  upper  leaf. 


THE    RIEMANN    SURFACE. 


143 


An  integral  is  called  closed  when  the  path  of  integration  reverts  to  the 
initial  point  in  the  same  sheet  from  which  it  started,  as  illustrated  in  the 
following  figures: 


Fig.  41. 

Cauchy's  Theorem  for  the  plane  is  also  true  of  the  Riemann  surface, 
viz.:  If  a  function  f(z)  within  a  portion  of  surf  ace  tha*  is  completely  bounded, 
the  boundaries  included,  is  everywhere  one-valued,  finite  and  continuous,  then 
the  integral  taken  over  the  boundaries  of  the  surface  in  such  a  way  that  it 
has  the  bounded  surface  always  to  the  left,  is  zero. 

ART.  122.  We  must  consider  more  closely  what  is  meant  by  the 
boundaries  of  a  portion  of  surface.  The  simplest  case  is  a  portion  of  surface 
as  shown  in  the  figure.  We  must  make  a  dis 
tinction  between  an  outer  edge  and  an  inner  edge. 
If  we  have  a  point  a  on  the  inner  edge  and  a  point 
b  on  the  outer  edge,  it  is  clear  that  we  cannot  go 
from  the  point  a  to  the  point  6  without  crossing 
the  boundary.  We  say  in  general  that  a  portion 
of  surface  is  completely  bounded  when  it  is  impos 
sible  to  go  from  a  point  on  the  inner  edge  to  a  point 
on  the  outer  edge  without  crossing  the  boundary. 

Consider  next  *  a  closed  curve  aft?  in  the  Riemann  surface.  We  may  go 
from  a  point  a  on  the  outer  edge  to  a  point  b  on  the  inner  edge  without 

crossing     the     curve 


Fig.  42. 


which  lies  wholly  in  the  up 
per  leaf.  Consequently  the 
curve  apf-  must  not  be  re 
garded  as  the  c  mplete 
boundary  of  a  portion  of 
surface.  But  if  we  also 

.p.    43  draw     a     congruent     curve 

otftf,  that  is,  one  imme 
diately  under  the  first  curve  and  in  the  lower  leaf  as  shown  in  Fig.  44, 
then  it  is  not  possible  to  go  from  the  point  a  to  the  point  b  without 


*  Cf.  Bobek,  loc.  tit.,  p.  155. 


144  THEORY   OF   ELLIPTIC    FUNCTIONS. 


crossing  one  or  the  other  of  the  two  curves  apj-  or  a'fi  f.  Hence  a/??- 
and  a'Pr'  together  form  the  complete  boundary  of  this  portion  of  sur 
face  of  the  Riemann  surface.  By  Cauchy's  Theorem  the  integral  taken 
over  /(z),  where  the  path  of  integration  extends  over  both  afir  and 
ci'fi'f'i  must  be  zero  if  the  direction  of  integration  is  taken  as  indicated 
above  and  if  f(z)  is  one-valued,  finite  and  continuous  within  and  on  the 
boundaries  of  this  surface. 


Upper  leaf 


Lower  leaf 


Fig.  44.  Fig.  45. 

To  prove  this  we  note  that  instead  of  taking  a$?  and  a'jf-f  as  the  paths 
of  integration  we  may  take  paths  which  lie  indefinitely  near  the  branch- 
cut  0,10,2,  this  one  branch-cut,  of  course,  lying  in  both  the  upper  and  the 
lower  leaf.  It  is  seen  that,  if  the  integration  is  taken  in  both  the  upper  and 
the  lower  leaf  (see  Fig.  45), 

/wdz  =    I  [p  +  qs]dz  =   I     2  qsdz  —I     2  qsdz  =  0, 
*J  *J  a-2.  <J  0,2 

the  elements  of  integration  taken  in  the  opposite  directions  over  /  pdz 
canceling  one  another. 

ART.  123.  If  a  one- valued  analytic  function  be  developed  in  the  2-plane 
in  the  form 


(z  -  a)A       (z  -  a)*"1  z  -  a 

where  P(z  —  a)  denotes  a  power  series  in  positive  integral  powers  of  z  —  a, 
then  we  know  that  the  residue  of  f(z)  with  respect  to  z  =  a  is 

61  =  Res  f(z), 

z  =  a 

the  quantity  bi  being  the  coefficient  of-     -  • 

The  same  definition  is  given  for  the  residue  of  a  function  of  position 
on  the  Riemann  surface,  provided  the  point  a  does  not  coincide  with  a 
branch-point. 

If,  however,  this  point  is  a  branch-point,  a\  say,  and  if  the  function 
becomes  infinite  at  this  point,  then  it  follows  from  above  that  the 
development  of  w  =  f(z)  in  the  neighborhood  of  this  point  is 

f(z)  =  (      6xW2  +  7 \l  n/2+  •  '  •+r-^LT|+,     bl    u  +  P{(s-oi)*}- 

(z-ai)A/2     (z  —  aiY^12  (*— 0l)*     (-2  -  fli)* 

Before  we  define  the  residue  here,  we  may  consider  a  theorem  which  gives 
the  residue  in  the  form  of  an  integral:    If  in  the  2-plane  we  draw  a  circle 


THE    RIEMA^N    SUKFACE.  145 

about  the  infinity  a  of  the  function  f(z)  and  if  f(z)  does  not  become  infinite 
on  any  other  point  within  or  on  the  circumference  of  this  circle,  then  is 


2~rf  J/(*)<fe  =  Res/(*)> 


where  the  integration  is  taken  over  the  circumference  of  the  circle.  We 
shall  also  retain  this  formula  as  the  definition  of  a  residue  on  the  Riemann 
surface  when  the  point  a  coincides  with  a  branch-point,  say  &i. 

The  integration  is  to  be  taken  over  a  complete  circuit  about  the  branch 
point,  that  is,  over  a  double  circle. 

We  may  write  under  the  sign  of  integration  instead  of  f(z)  the  power 
series  by  which  it  is  represented.  The  general  term  is 

(z  -  arfdz, 

where  the  integration  is  over  the  double  circle. 

Suppose  that  r  is  the  radius  of  the  double  circle,  so  that 

z  —  a  i  =  re^, 
and  consequently  also 

^d'ble-circle  ^0  Jo 

This  integral  is  always  zero,  except  when  1  +  -  =  0.     In  this  latter  case 


'd'ble-circle 

It  follows  that 


I  (z  -  a{)'2dz  =  i  I      d(f> 

^d'ble-circle  ^0 

Res/(z)  -  JL   Cf(z)dz  =  -LT 

z  =  di  *  '*1  ^d'ble-Hrrle       ^  '*l 


d'ble-circle 

where  62  is  the  coefficient  of  (z  —  cti)~*,  since  k  =  —  2.     We  thus  have 
finall>'  Res/(z)  =2b2; 

z=ai 

or,  the  residue  with  respect  to  a  branch-point  is  equal  to  DOUBLE  the  coefficient 
of  (z  —  ai)"1  in  the  development  of  the  function  in  powers  of  (z  —  &i)*. 

ART.  124.  Suppose  that  a  portion  of  surface  is  given  which  is  completely 
bounded  by  certain  curves.  At  isolated  points  of  this  surface  suppose  that 
the  function  becomes  infinite.  We  draw  around  these  points  small  cir 
cles,  simple  if  they  are  not  branch-points,  and  double  when  they  are  branch 
points.  The  interior  of  these  circles  we  no  longer  count  as  belonging  to 
the  surface.  In  this  manner  we  derive  a  new  portion  of  surface  which  is 
completely  bounded  on  the  one  hand  by  the  original  curves  and  on  the 
other  by  the  small  circles.  The  integral  taken  over  the  boundaries  of 
this  new  portion  of  surface  is  zero,  since  the  function  is  everywhere  finite 


146  THEORY   OF   ELLIPTIC   FUNCTIONS. 

within  this  surface,  boundaries  included.     The  integration  is  to  be  so 
taken  that  the  interior  of  the  portion  of  surface  is  always  to  the  left.     If 

the  direction  of  integration  taken  over  the 
small  circles  is  changed  so  that  the  interiors 
of  these  circles  lie  to  the  left  of  the  integra 
tion,  then  the  signs  of  the  corresponding 
integrals  must  be  changed,  and  we  have  the 
following  theorem:  The  integral  over  the  com 
plete  boundaries  of  the  original  portion  of 
surface  is  equal  to  the  sum  of  the  integrals  over 
the  circles  (or  double-circles)  which  are  drawn 
around  the  infinities  (poles). 

But  on  the  other  hand  each  of  the  inte 
grals  around  one  of  the  circles  is  equal  to  the  residue  of  the  function 
with  respect  to  the  infinities  in  question  multiplied  by  2  id.  We  have 
therefore  the  theorem  :  //  a  function  within  a  completely  bounded  portion 
of  surface,  boundaries  included,  is  everywhere  one-valued  and  discontinuous 
only  at  isolated  points,  then  the  integral  multiplied  by  1/2  ni  and  taken  over 
the  complete  boundary  of  this  surface  is  equal  to  the  sum  of  the  residues  of 
the  function  with  respect  to  all  the  points  of  discontinuity  within  the  portion 
of  surface. 

ART.  125.     We  saw  that  any  one-  valued  function  of  position  on  the 
Riemann  surface  s  =  \/R(z)  was  of  the  form  * 

w  =  p  +  qs, 

where  p  and  q  are  rational  functions  of  z  and  where  s  =  v72(z). 
It  follows  that 


w  dz  p  +  qs  p  -f  qs 

If  the  numerator  and  the  denominator  of  the  right-hand  side  of  this 
expression  are  multiplied  by  p  —  qs,  we  have 


w 


where  P  and  Q  are  rational  functions  of  z. 

It  is  thus  seen  that  the  logarithmic  derivative  of  w  =  p  +  qs  is  a  rational 
function  of  z  and  s  and  indeed  of  the  same  form  as  is  w  itself. 

The  logarithmic  derivative  becomes  infinite  at  the  points  where  w 
vanishes  and  at  the  points  where  w  becomes  infinite. 

*  See  Riemann,  Werke,  p.  111. 


THE   BIEMAXX    SURFACE.  147 

If  /j.  is  the  order  of  the  zero  of  the  function  w  at  the  point  a,  then  in  the 
neighborhood  of  a 

d-^^  =  -*—  +  P(z  -  a)         [see  Art.  4]; 
dz          z  —  a 

and  if  ^  is  the  order  of  infinity  of  the  function  w  at  the  point  /?,  then  in 
the  neighborhood  of  p 


dz  z  - 

It  follows  that 


and 

z=3      dz 

The  above  discussion  is  true  when  a  and  /9  are  not  branch-points. 

If  a  is  a  branch-point,  say  a1?  and  if  w  becomes  zero  at  this  point,  then 
in  the  neighborhood  of  this  point  we  have 

p 
w  =(z  -  ai)2[#0  +  0i  0  -  «i)*  +  92(2  -  ai)§+  •  -  •  1 

and  consequently 

log  w  =  |  log  (z  -  ai)+  log[£o  +  0i  (2  -  «i)*  +  •••]• 

It  follows  that 

& 
^  -  -^—  +  flog  [0o  +  ffl(z  -  fll)*  +  -  .  .  1 

C?2  2  —   Ox          C?2 

Since  the  logarithmic  expression  does  not  become  infinite  for  z  =  a\, 
it  maj'  be  developed  in  the  form 

log  [0o  +  g\(*-  ai)*  +  '  •  '1  -*•>*!(«  -«i)*  +  "  '  '» 

and  consequently 

ff 


dz  z  —  ai        2 

We  therefore  have  (cf.  Art.  120) 

z  =  ai         dz  2 

If  on  the  other  hand  a!  is  an  infinity  of  the  Ath  order  of  w,  then  is 

-P      d  log  w  3 

Kes  — * —  =  —  X. 


148  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  126.     We  shall  now  apply  Cauchy's  Theorem  to  the  function 


As  the  portion  of  surface  over  whose  boundaries  the  integration  is  to  be 
taken  we  shall  choose  a  region  which  contains  all  the  infinities  of  the 
function  P  +  Qs. 

In  order  to  have  such  a  surface,  we  construct  in  the  Riemann  surface 
a  very  small  circle  which  does  not  contain  any  of  the  infinities  of  P  +  Qs. 
The  rest  of  the  Riemann  surface,  that  is,  the  entire  Riemann  surface  except 
ing  the  small  circle,  will  then  contain  all  the  infinities  of  P  +  Qs.  The 
point  at  infinity  may  be  one  of  these  infinities.  In  the  latter  case  we  make 

the  substitution  z  =  -.  The  function  P  +  Qs  becomes  by  this  substitu 
tion,  say 


P  +  Qs  =  P,  (t)  +  Q0(t)  Y/I  -  «,)  (I  -  a2]  (I  -  a,}  (\  -  a4] 

'      \t  /    \t  /   \t  /  \t  I 


-  Pi(t)+Qi(t)      (l  -  ait)(l  -  a2t)(l  -  a3t)(l  -  a40, 
where 


The  functions  Pi(f)  and  Qi(t)  are  rational  functions  of  t;  and  in  the  £-plane 
the  origin  is  now  an  infinity.  The  other  infinities  in  the  old  Riemann  sur 
face  remain  at  finite  distances  from  the  origin  on  the  new  Riemann  surface, 
whose  branch-points  are  the  reciprocal  of  those  in  the  old  Riemann  surface.* 

We  thus  have  no  trouble  in  computing  the  order  of  the  infinity  at  the 
point  infinity. 

The  boundary  of  the  region  is  evidently  that  of  the  small  circle,  and  the 
integration  is  to  be  taken  so  that  the  region  without  the  circle  lies  to  the 
left. 

After  the  theorem  of  Art.  92,  when  we  remain  on  the  original  Riemann 
surface 


where  the  integration  is  taken  so  that  the  bounded  region  is  on  the  left, 
that  is,  so  that  the  interior  of  the  small  circle  is  on  the  right. 

Noting  that  the  integral  taken  over  the  boundary  of  this  small  circle, 
within  which  there  is  no  infinity  of  the  function,  is  zero,  it  is  seen  that 

2  Res  (P  +  Qs)  =  0, 

where  the  summation  extends  over  all  the  infinities  of  P  +  Qs. 
*  Cf.  Neumann,  loc.  cit.,  p.  111. 


THE   RIEAIANX    SURFACE. 


149 


These  residues  fall  into  two  groups:  those  of  the  one  group  have  refer 


ence  to  the  infinities  of  the  function 


,  which  exist  through  the  van- 


ishing  of  w,  while  those  of  the  other  group  refer  to  the  infinities  of  -  , 
which  are  also  the  infinities  of  w. 

If  by  H//  we  denote  the  sum  of  the  orders  of  the  zeros  and  by  2x  the  sum 
of  the  orders  of  the  infinities  of  w,  then  for  P  +  Qs  the  sum  of  the  residues 
of  the  first  group  is  S/£,  while  —  Sx"  is  the  sum  of  the  residues  of  the  second 
group. 

It  follows  at  once  that 


2  Res  (P  +  Qs}  = 


-  SJ  =  0, 


or 


It  has  thus  been  shown  that  the  sum  of  the  orders  of  the  zeros  of  w  is 
equal  to  the  sum  of  the  orders  of  its  infinities;  or,  in  other  words,  the  function 
w  becomes  as  often  zero  as  it  does  infinity  in  the  Riemann  surface,  if  a  zero  of 
the  ath  order  is  counted  u-ply  and  an  infinity  of  the  Ath  order  is  counted  A-ply. 

ART.  127.     Suppose  that  k  is  an  arbitrary  constant  and  write 

p  +  qs  =  k. 


The  function  p  +  qs  —  k  is  a  rational  function  in  z  and  s. 
infinite  as  often  as  p  +  qs  is  infinite,  and  since  the  relation 


It  becomes 


is  true  also  here,  it  becomes  zero  as  often  as  p  +  qs  becomes  zero. 
We  thus  have  the  following  theorem:     The  equation 

p  +  qs  =  k 

has  in  the  Riemann  surface  as  many  solutions  as  p  -f  qs  has  infinities. 
Hence  also  the  function  p  +  qs  takes  every  value  in  the  Riemann  surface 
an  equal  number  of  times. 

ART.  128.  We  have  often  employed  the 
term  "  complete  boundary "  and  have  in 
particular  considered  this  expression  in  Art. 
122.  We  shall  again  emphasize  the  fact 
that  it  is  of  extreme  importance  to  under 
stand  the  full  significance  of  this  term.  If 
from  a  portion  of  surface  A  a  piece  is  cut 
out,  for  example  a  circle  around  a  point  of 
discontinuity,  then  in  this  new  portion  of 
surface  every  closed  curve  no  longer  forms 
a  complete  boundary.  If  P  is  the  small 

circle  that  has  been  cut  out  of  A,  then  the  closed  curve  B  no  longer 
forms  a  complete  boundary,  since  B  and  C  together  constitute  this  corn- 


Fig.  4; 


150 


THEOKY   OF   ELLIPTIC    FUNCTIONS. 


Fig.  48. 


plete  boundary.     If  from  any  portion  of  the  surface  A  we  cut  out  a  circle 

and  join  this  circle  with  the  original  boundary  by  means  of  a  cross-cut, 
it  is  then  impossible  to  draw  a  closed  curve  in  A 
which  does  not  form  the  complete  boundary  of  a 
portion  of  surface,  so  long  as  we  do  not  cross  the 
cross-cut. 

Every  surface  which  has  the  property  that  every 
closed  curve  drawn  in  it  is  the  complete  boundary 
of  a  portion  of  surface,  is  called  a  simply  connected 
surface.* 

The  Riemann  surface  on  which  the  function  w  = 
p  +  qs  is  represented  is  not  a  simply  connected  one. 

We  may,  however,  as  shown  in  the  figure,  easily  transform  it  into  a  simply 

connected  surface  by  drawing  the  two  canals  a  and  b.     We  note  that  one- 

half  of  the  canal  b  lies  in  the  lower  leaf.     These  canals  cannot  be  crossed 

by  going  from  one  leaf  into 

the  other  as  is  the  case  with 

the  canals  a^2  and  a3a4. 
The  Riemann  surface  con 

taining  the  two  canals  a  and 

b  we  denote  by  T'.     The  sur 

face    which   does    not    have 

these   canals  is  denoted   by 

T.      The  surface  Tf  is  said 

to  be  of  order  f  unity.      We 

note   that   two  canals  or  cross-cuts  were  necessary  to  make  it  simply 

connected.     One  may  easily  be  convinced  by  trial   that   every  closed 

curve  in  T'  forms  the  complete  boundary  of  a  portion  of  surface,  so  long 

as  the  curve  does  not  cross  the  canals  a  and  b. 

ART.  129.     Anticipating  some  of  the  more  complicated  results  of  the 

next  Chapter,  we  may  consider  here  the  simpler  case  of  the  function 

s2  -  r(z), 

where  r(z)  =  A(z  —  0,1)  (z  —  a2).     The  associated  Riemann  surface  con 
sists  of  two  leaves  connected  along  the  canal  ai«2- 
The  integrals  ~     . 

P  =  r^, 

J  Vr(z) 

*  Cf.  Neumann,  loc.  cit.,  p.  146. 

f  In  general,  if  TV  denotes  the  number  of  branch-points  belonging  to  any  function, 
n  the  number  of  leaves  in  the  associated  Riemann  surface,  and  p  the  class  or  order 
of  the  Riemann  surface,  then  (see  Forsyth,  Theory  of  Functions,  p.  356)  TV  =  2  p 
+  2  n  —  2.  (Cf.  Riemann,  Werke,  p.  114.)  The  name  deficiency  was  introduced  by 
Cayley,  On  the  Transformation  of  Plane  Curves.  1865.  The  deficiency  of  a  curve  is  the 
class  or  order  of  the  Riemann  surface  associated  with  its  equation;  that  is,  t/2=  R(x)  is 
a  curve  of  deficiency  unity,  if  s2=  R(z)  is  a  Riemann  surface  of  order  unity. 


49 


THE   RIEMANN    SURFACE. 


151 


where  the  paths  of  integration  are  taken  over  the  two  curves  (1)  and  (2), 
are  equal  since  the  function      ,  —  -  is  one-valued  finite  and  continuous 

for  all  points  of  the  surface  between  these 
two  curves. 

If  we  let  the  path  of  integration  (1) 
approach  indefinitely  near  the_canal  a,ia2, 
then,  since  the  values  of  Vr(j)  on  the 
right  and  left  banks  of  this  canal  have 
contrary  signs,  we  have 


dz 


where  in  the  last  integral  the  integration  is  taken  along  the  upper  leaf  and 
the  left  bank. 

It  follows  that  P  is  different  from  zero  and  consequently  also  the  integral 
taken  over  a  curve  such  as  (2)  is  not  zero. 

This  two-leaved  Riemann  surface  T  we  next  cut  by  a  canal  so  that  the 
integral 


r  dz 

u  =    I  — = 

J  Vr(z} 


Vr(z) 

will  be  a  one- valued  function  of  position  in  the  surface  where  the  cut 
has  been  made.  This  integral  will  then  be  independent  of  the  path 
of  integration,  which  we  have  just  shown  by  going  around  the  canal  a,ia2 
is  not  the  case  in  the  Riemann  surface  before  the  cut  has  been  made. 

From  a  point  C  on  the  upper  bank  of 
the  canal  we  draw  a  line  CA  which  goes 
off  towards  infinity  and  this  line  is  indefi 
nitely  continued  from  C  in  the  other  direc 
tion  in  the  lower  leaf.  We  thus  form  a 
cut  or  canal  AB  which  is  not  to  be  crossed. 
The  surface  with  this  new  canal  we  call  T'. 
From  the  figure  it  is  seen  that  we  may 
go  from  any  point  a  on  the  bank  of  the 
canal  AB  to  a  point  ft  immediately  oppo- 
/B  site  on  the  other  bank  without  crossing 

Fig  51  either  the  canal  AB  or  the  canal  a,id2,  but 

it  is  impossible  to  make  a  circuit  around 

the  canal  a\a2  or  around  either  of  the  branch-points  a!  or  a2  without 
crossing  one  of  these  canals.  It  follows  that  the  above  integral  in  T'  is 
one-valued. 


152 


THEORY    OF   ELLIPTIC    FUNCTIONS. 


ART.  130.     Next  let 


and  let 


u(z,  s)= 


u(z,s)= 


dz 


Vr(z)' 
dz 


where  the  path  of  integration  is  in  Tf; 


where  the  path  is  in  T. 


»,sGVr(z) 

The  integration  in  both  cases  is  always  counted  from  a  fixed  point  20, 
s0,  which   as  a  rule    may    be   arbitrarily  taken,   but   when   once   taken 

must  be  retained  as  the  lower  limit  for 
all  the  integrals  that  come  under  the  dis 
cussion. 

We  know  that  if  the  function  Vr(z)  is  one- 
valued,  finite  and  continuous  within  the 
area  situated  within  the  two  curves  (1) 
and  (2)  of  the  figure, 


j%2j  $2  j^2t  $2  §2\f  ^1  j^2r  $2 


dz 


Fig.  52. 
It  follows  that  in  T' 


o,  S2  (*Z2,S2 

=     I 

zi,  Si  *Jz0,  S0 


(1)  (2) 

where  the  integrand 

Vr(z) 
stood  with  every  integral. 


=  t*(«2,  S2)   - 


-  is  to  be  under 


Next  take  the  integral  from  z0,  s0  to  2,  s  in  T  where  there  being  no  canal 
AB  we  go  by  the  way  of  the  two  points  A  and  p. 
We  have 

(*(>         r*        f*z,  s 
u(z,  s)  =    /       +    /      +    /       , 

i/JBO,  S0          *J p  *J  * 

where  the  distance  between  ^  and 
p  being  indefinitely  small  the  middle 
integral  on  the  right  may  be  neg- 
lected.  But  from  above 


and 


I 


=  u(z,  s)  -  u(X), 


1  ig.  53. 


where  both  of  these  integrals  are  in  the  Tf  surface.  From  this  it  is  seen 
that  u(Zj  s)  =  u(z,  s)  +  u(p)  -  u(X), 

where  u(p)  and  7i(X)  are  the  integrals  from  z0,  s0  to  p  and  from  ZQ,  s0  to 
X  in  the  T'  surface,  the  path  of  integration  being  taken  in  any  manner  so 
long  as  neither  of  the  canals  a^2  and'AZ?  is  crossed. 


THE    RIE3IAXX    SURFACE. 


153 


On  the  other  hand, 


Vr(z) 
or,  from  the  figure, 


and  since 


we  have 


where   the  integration  of  the  last  integral 
is  taken  in  the  upper  leaf   and    the  lower 
bank  of  the  canal  aja2  . 
We  have  finally 

u(z,  s)=u(z,  s)+P, 


r~ 


Fig.  54. 


where  P  is  a  quantity  which  does  not  depend  upon  the  path  ZQ,  s0  to  zit  si. 

The  quantity  P  is  called  the  modulus  of  periodicity. 

If  the  path  of  integration  is  taken  so  that  we  pass  from  the  right  to  the 
left  bank  of  the  canal  AB,  then  is 

u(z,  s)  =  u(z,  s)  -  P. 

The  integral  in  T  differs  from  the  integral  in  T'  only  by  a  positive  or 
negative  multiple  of  P,  this  multiple  depending  upon  the  number  of  times 
and  the  direction  the  canal  AB  has  been  crossed  [see  Durege.  Ellintische 
Functionen  (2d  ed.),  p.  370].  \ 


EXAMPLE 


/j 
— 
\T\  - 


VI  -z2 


REALMS  OF  RATIONALITY. 

ART.  131.  Let  z  be  a  complex  variable  which  may  take  all  real  or 
complex,  finite  and  infinite  values.  Consider  the  collectivity  of  all  rational 
functions  of  z  writh  arbitrary  constant  real  or  complex  coefficients.  These 
functions  form  a  closed  realm,  the  individual  functions  of  which  repeat 
themselves  through  the  processes  of  addition,  subtraction,  multiplication 
and  division,  since  clearly  the  sum,  the  difference,  the  product,  and  the 
quotient  of  two  or  more  rational  functions  is  a  rational  function  and  con 
sequently  an  individual  of  the  realm. 


154  THEORY   OF   ELLIPTIC    FUNCTIONS. 

This  realm  of  rationality  we  shall  denote  by  (z).  Consider  next  the  one- 
valued  functions  on  the  fixed  Riemann  surface.  If  we  denote  any  such 
function  by  wi  =  pi  +  qis  and  any  other  such  function  by  w2  =  P2  +  ?2«, 
then  the  sum,  difference,  product  and  quotient  of  the  two  functions  w\  and 
W2  are  functions  of  the  form 

w  =  p  +  qs. 

It  is  evident  that  if  we  add  (or  adjoin)  the  algebraic  quantity  s  to 
the  realm  (z),  we  will  have  another  realm  (z,  s),  the  individual  functions 
or  elements  of  which  repeat  themselves  through  the  processes  of  addi 
tion,  subtraction,  multiplication  and  division.  This'  realm  we  shall  call 
the  elliptic  realm.  It  includes  the  former  realm.  We  note  that  every 
element  of  this  realm  is  a  one-valued  function  of  position  on  the  fixed 
Riemann  surface.  In  the  present  Chapter  we  have  proved  that  every 
element  of  the  realm  (z,  s)  takes  every  arbitrary  value  that  it  can  take 
an  equal  number  of  times.  It  also  follows  that  within  this  elliptic  realm 
there  does  not  exist  an  element  that  becomes  infinite  of  the  first  order 
at  only  one  point  of  the  Riemann  surface.  This  latter  statement  is  left 
as  an  exercise  (see  Thomae,  Functionen  einer  complexen  Verdnderlichen, 
p.  94). 


CHAPTER   VII 
THE  PROBLEM  OF  INVERSION 

ARTICLE  132.  We  have  seen  (Chapter  V)  that  every  one-valued  doubly 
periodic  function  of  the  second  order  which  has  no  essential  singularity 
in  the  finite  portion  of  the  plane,  or  Riemann  surface,  satisfies  a  certain 
differential  equation  in  which  the  independent  variable  does  not  explicitly 
appear.  This  equation  may  be  written 


du  ' 

where  p(z)  is  an  integral  function  of  at  most  the  second  degree  and  R(z) 
is  an  integral  function  of  the  fourth  degree.  We  saw  in  the  preceding 
Chapter  that  p(z)  +  \/R(z)  is  a  one-valued  function  of  position  on  the 
fixed  Riemann  surface.  WTe  are  thus  led  to  the  study  of  the  integral 

dz 


p(z)  +  VR(z) 

As  the  lower  limit  of  this  integral  we  take  any  point  z0  of  the  Riemann 
surface,  at  which  s  has  the  value  s0=  +Vfi(«o).     Throughout  the  whole 
discussion  this  point  ZQ,  SQ  will  be  taken  as  the  initial  point.     The  integral 
is  taken  along  any  path  of  integration  to  the  point  z,  s. 
It  follows  then  that 


is  a  definite  function  of  the  upper  limit,  a  function  which  is  dependent  upon 
the  path  of  integration. 

We  may  also  consider  the  upper  limit  z  as  a  function  of  u;  and  we  shall 
now  raise  the  question:  Under  what  conditions  is  the  upper  limit  z  a  one- 
valued  function  of  uf 

It  is  possible  that  the  point  z,  s  lies  in  the  neighborhood  of  a  branch 
point  a  i,  say. 

We  then  have  the  following  development: 


and  consequently 


We  thus  have  a  series  which  proceeds  in  ascending  powers  of  (z 

155 


156  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  133.     Suppose  that  p(ai)  does  not  vanish. 

i 

We  may  then  develop  • in  integral  powers  of  (z  —  c^)*  in 

the  form  P.W  +  v  /t(2) 

'-  ci(z-  01)*  +  c2(z- 


p(z)  +  VR(z) 
If  we  put  p^  ^ 

Jz0,Sop(z)  +  VR(z) 
it  is  seen  that 


=  a, 


/*«,«  ^p-  /»0i  /7?  /'Z.a 

u  =  I      az  I      az  +  /      

Jz»,  s0  p  (z)   +  VR  (z)        Jz0,  So  p  (Z)   +  VR  (z)        t/o,       p  (z) 


dz 


=  a  + 


r*> 

*J  a\ 


p(z)  +  VR(z) 

We  have  here  assumed  that  the  point  z,  s  has  been  so  chosen  that  there  is 
no  point  of  discontinuity  of  the  integrand  within  the  triangle 
swso  It  follows  that 

u  —  a  = 

z's  By  hypothesis  the  point  z,  s  lies  in  the  neighbor 
hood  of  a\j  that  is,  on  the  inside  of  a  circle  within 
which  the  series  developed  above  is  convergent.  We 


ttl        F.     55          2    may  therefore  integrate  this  series  and  have 


u  —  a  =   I       ] h  ci(z  —  ai)^+  c2(z  —  ^j)l  -f  •   •   •  >  d(z  — 

Jai     lp(&i)  } 

=  z-  al  +2c(z 

p(a1)        3°] 
If  we  put  z  —  a  i  =  t2,  we  have 


It  follows  that 


or  (u  —  a)*  = 

where  of  course  the  quantities  ci,  c2,  .  .  .  ,  /2,  etc.,  are  constants.     By 
the  reversion  of  this  series  we  have 

But  since  t2  =  z  —  a\,  it  is  seen  that  z  is  two-valued  and  not  one-valued  in 
the  neighborhood  of  u  =  a. 


THE   PROBLEM   OF   INVERSION.' 


157 


ART.  134.     If  ;Xai)  =  0>  the  above  development  becomes 
1 


p(z)  +  VR(z) 
We  then  have 

u  —  a  =   I  [e-i(z  — 
=  2e-i(z  -  a 


e-i(z  - 


e0 


e0(z  -  01)*  + 


}d(z-al) 


From  this  we  conclude  that  t  is  developable  in  positive  integral  powers 
of  u  —  a  and  consequently  is  one-valued  in  the  neighborhood  of  u  =  a. 
It  follows  also  that  z  is  one-valued  in  the  neighborhood  of  this  point. 

Hence  in  order  that  z  be  a  one-valued  function  of  u,  it  is  necessary 
that  p(ai)  =  0.     In  the  same  way  it  may  be  shown  that  p(a2)  =  0  = 


On  the  other  hand,  p(z)  is  an  integral  algebraic  function  of  at  most  the 
second  degree  in  z.  Such  a  function  cannot  vanish  at  more  than  two 
points  without  being  identically  zero.  It  follows  that  p(z)  =  0.  We 
therefore  have  the  theorem:  In  order  that  z  be  a  one-valued  function  of  u, 
it  is  necessary  that  p(z)  =  0  and  consequently  also  that 

du 

ART.  135.     The  last  investigation  would  be  true  even  if 

dz 


N*P(Z) 

were  infinite.     We  may  prove,  howrever,  as  follows  that  this  integral  is 
never  infinite. 


We  saw  above  that 


1 


p(z) 


-==  is  developable  in  a   power  series 


which  is  convergent  within  a  certain 
circle.  Let  this  circle  cut  the  path  of 
integration  at  the  point  zf,  s'.  We 
then  have 

dz  C2''*'          dz 


«/* 


,*  p(z) 
dz 


Uo 


Fig.  56. 


',*'p(z)  +  VR(z) 

The  first  integral  on  the  right  is  finite, 
since  it  does  not  become  infinite  for 
any  value  between  z0,  «o  and  zf,  s';  while  the  second  integral,  as  shown 
above,  may  be  expressed  through  the  series 
[_2  6-1(2  —  ai)*  +  CQ(Z  — 


158 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


This  series  is  finite  for  the  values  z',  s'  and  a\.     It  follows  therefore  that 

dz 


/* 
= 

*J  Z 


»«p(z)  +  VR(z) 

has  a  finite  value  even  when  p(ai)  =  0,  and  at  the  same  time  it  has  been 
shown  that  the  integral 

dz 


is  finite  when  the  upper  limit  is  a  branch-point. 

ART.  136.     We  may  now  confine  ourselves  to  the  consideration  of  the 

integral  /^s     d 

u  =  I  • 

Jzo,soVR(z) 

This  integral  is  called  an  elliptic  integral  of  the  first  kind.  We  have  seen 
that  the  integral  u  remains  finite  when  the  upper  limit  coincides  with  a 
branch-point.  We  shall  next  see  that  this  integral  remains  finite  when 
the  path  of  integration  goes  into  infinity. 

In  one  of  the  leaves  of  the  Riemann  surface,  for  example  the  upper, 
draw  a  circle  with  the  origin  as  center  which  includes  all  the  branch-points. 
On  the  outside  of  this  circle  the  quantity  \/R(z)  and  consequently  also 

— is  one-valued;  for  if  we  make  a  closed  circuit  without  this  circle 

VR(z) 

it  includes  either  none  or  all  the  branch-points  and  consequently  — 

VR(z) 
does  not  change  its  value. 

We  have 
1 

VlR(zj  ~  < 

Since  —  >    —,    —,    —  are  proper  fractions  for  all 

z       z       z       z 

values  of  z  without  this  circle,  each  of  the  above 
factors  is  developable  in  positive  integral  powers 

of  -,  so  that 

z 


Fig.  57. 


VRCz) 


which  series  is  convergent  for  all  values  of  z  without  the  circle. 

Let  z',  sf  be  the  point  where  the  path  of  integration  starting  from  the 
point  z0,  s0  and  leading  to  infinity,  cuts  the  circle. 

We  have 


THE   PROBLEM   OF   INVERSION.  159 

We  have  seen  that  the  first  integral  on  the  right  is  always  finite,  whether 
the  path  of  integration  goes  through  a  branch-point  or  not.  For  the 
second  integral  we  have 

' 


r*-_r  &»+$+...  1* 

J*.*vB(d    Js.s'Lz2    «3          J 

T 
>. 


2*2 

an  expression  which  is  finite  for  both  the  upper  and  the  lower  limit.     We 
have  thus  shown  that  the  integral 

:-s     dz 


f 

Jz«, 


VR(z) 

is  finite  everywhere,  even  when  the  upper  limit  is  indefinitely  large  or  if  it 
coincides  with  one  of  the  branch-points. 

ART.  137.  We  represent  by  T'  the  Riemann  surface  of  Art.  128  in 
which  the  canals  a  and  b  have  been  drawn.  We  noted  that  any  closed 
curve  on  this  surface  formed  the  complete  boundary  of  a  portion  of  sur 
face.  If  on  this  surface  the  curve  C  includes  one  or  several  branch-points, 
for  example  ai,  we  isolate  them  by  means  of  small  double  circles.  If 
K  denotes  the  double  circle  about  ai,  and  if  the  curve  C  includes  only  one 
such  branch-point,  then  by  Cauchy's  Theorem  we  have 


C  4*  +   C  4*  =  o,  where  s  = 
Jc  s      JK  8 


c 

Note  that  in  this  second  integral  the  integration  is  over  two  circles  lying 
directly  the  one  over  the  other  in  the  two  leaves  of  the  Riemann  surface. 
In  these  two  leaves  the  quantity  s  has  opposite  signs,  while  at  points  the 
one  over  the  other  the  absolute  values  of  s  and  z  are  equal.  It  follows 

that  in  the  integral   /    —  the  elements  of  integration  cancel  in  pairs,  so 
JK  s  n.  jz 

that  this  integral  is  zero.     We  have  thus  shown  that  the  integral  /   - 

Jc  s 
taken  over  any  closed  curve  in  T'  is  zero. 

If  in  T'  we  draw  any  two  curves  (1)  and  (2)  between 
the  points  ZQ,  s0  and  2,  s,  without  crossing  either  of  the 
canals  a  or  b,  the  two  curves  will  form  a  closed  curve,  and 
from  what  we  have  just  seen  (1) 


dz        r^'^dz  = 

8  1/2,  S  8 

(1)  (2) 

C*>sdz  =   Czs  dz 

J  ZQ,  SQ    S  J  ZQ,  So     S 


or 

a>"  (2)~  Fig.  58. 

the  numbers  in  parentheses  under  the  integral  signs  denoting  the  paths 
along  which  the  integration  has  been  taken. 


160  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Hence  if  we  write 


where  the  dash  over  u  signifies  that  the  integration  is  to  be  taken  in  the 
Riemann  surface  T' ',  in  which  the  canals  a  and  b  are  not  to  be  crossed,  it 
follows  from  above  that  u  (z,  s)  is  entirely  independent  of  the  path  of  inte 
gration.  It  follows  also  that  the  integral  u(z,  s)  is  a  one-valued  definite 
function  of  the  upper  limit 

ART.  138.     We  shall  consider  next  the  integral 

U(Z,   8)  = 

where  the  path  of  integration  is  taken  in  the  Riemann  surface  T,  which 
does  not  contain  the  canals  a  and  b.  We  shall  show  that  here  the  integral 

u(z,  s)  is  not  a  one-valued 
function  of  the  upper  limit 
z,  s,  but  -depends  upon  the 
path  of  integration. 

In  the  T-surface  the  inte 
gral  corresponding  to  u(z,  s) 


s 


u(z,  s)  = 


dz 


dzm 

s 

The  points  p  and  X  are 
supposed  to  lie  indefinitely 
near  each  other,  so  that  the  middle  integral  to  the  right  is  zero.  We  con 
sequently  have 
(A)  u(z,  s) 


Fig.  59. 


We  have  seen  that  in  the  Riemann  surface  Tf  every  integral  is  indepen 
dent  of  the  path  of  integration.     We  note  that  (see  Art.  130) 


_f  =u  (z2,  s2)  —  u(zi,  si). 


Returning  to  the  equation  (A),  it  is  seen  that  neither  of  the  canals  a  or 
is  crossed  between  zQ,  SQ  and  p,  so  that 


f"  4*  = 

JZQ,SQ     $ 


n 


THE   PROBLEM   OF   INVERSION.  161 

Further-,  there  is  no  canal  between  A  and  z,  s.  It  follows  from  what  we 
have  just  shown  that 

'°dz  =  u(g  s)  -  u(X)  in  T', 

s 

where  we  go  in  T'  from  ZQ,  s0  to  z,  s  by  crossing  the  canals  a3  a4  and 
a  i  a2  as  shown  in  the  figure.  We  have  to  make  the  same  crossings  to  go 
from  zQ)  SQ  to  L  We  therefore  have  from  the  equation  (A) 

u(z,  s)  =  u(z,  s)  +  u(p)  —  u(X). 

If  the  canal  a  had  been  crossed  at  any  other  point  pi,  AI  instead  of  at  p,A, 
we  would  have  had 

u(z,  s)  =  u(z,  s)  +  u(pi)  — 

Consider  the  difference 


The  points  p  and  pi  are  both  on  the  same  side  of  the  canal  a,  while  the 
point  X  and  Xi  are  both  on  the  opposite  bank. 
It  is  seen  that 


u(p)  -u(Pl)  -  f 
JP1 


s 


and  u(X)  -  u(^)  =        —  in 


s 


where  the  path  of  integration  in  T  may  be  quite  arbitrary,  provided 
only  it  does  not  cross  the  canals  a  and  &.  We  may  therefore  take  the  path 
of  integration  from  p  to  pi  indefinitely  near  the  right  bank  of  the  canal, 
while  the  path  from  X  to  ^  is  taken  indefinitely  near  the  left  bank.  Since 
these  two  paths  differ  from  each  other  by  an  infinitesimal  quantity,  the 
integrals  over  them  are  equal.  It  follows  then  that 

{u(p)  -  u(^\-\u(io1)  -  M(^i)}  =  0, 

and  consequently  u(p)  —  u(X)  has  the  same  value  at  whatever  point  the 
crossing  has  taken  place. 

ART.  139.     If  we  cross  the  canal  a  from  z0,  s0  to  z,  s  in  the  opposite 
direction  from  that  gone  over  in  the  previous  case,  we  have 


Cz>sriz     r^  d?     rz,*fj? 

I        *=   I       ™.  +  I      «£  (i 

*SZQ,SQ    S  JZO,SQ     S  *)  p  S 


-  u(X)  +  Ti(z,  s)  -  u(p)  (in  T'), 
=  u(z,  s)  +  u(X)  -  u(p). 


162 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


We    note   that   in  T'  we  must  go  from   z0,  s0  to  the    canal  joining  ax 
and  a2  and  after  crossing  this  canal  into  the  lower  leaf  come  out  again 

into  the  upper  leaf  by  crossing 
the  canal  a3a4  and  then  pro 
ceed  to  z,  s.  We  thus  see  that 
when  we  cross  the  canal  a  in  the 
opposite  direction  to  that  fol 
lowed  in  the  previous  article  we 
have  to  subtract  the  quantity 
u(p)—u(X)  from  u(z,  s). 

If  the  canal  a  is  crossed  // 
times  in  the  first  direction  and 
v  times  in  the  second  direction, 


Fig.  60. 


we  will  have 
u(z,  s)  =  u(z,  s) 


We  have  precisely  the  same  result  if  we  cross  the  canal  b.  Of  course,  the 
constant  u(p)  —  u(X)  is  different  here  from  what  it  was  in  the  previous 
case  when  we  crossed  the  canal  a. 

We  shall  write 

for  the  canal  a  :  u(X)  —  u(p)  =  A, 
for  the  canal  b  :  u(p)  —  u(X)  =  B. 

We  therefore  have  in  general 

u(z,  s)  =  u(z,  s)  +  mA  +  nB, 

where  m  and  n  are  positive  or  negative  integers  and  where  u(z,  s)  is  the 
integral  in  which  the  path  of  integration  is  free,  u(z,  s)  being  the  integral 
in  the  Riemann  surface  T',  in  which  the  canals  a  and  b  cannot  be  crossed. 
The  quantities  A  and  B  are  called  the  Moduli  of  Periodicity. 

ART.  140.     We  have  seen  that  if  a  and  6  are  two  quantities  whose  quo 

tient  is  not  real  and  if  the  coefficient  of  i  in  the  complex  quantity  -  is 

positive,  we  may  determine  a  function  $(u)  which  satisfies  the  two  func 
tional  equations 

$(u  +  a)  =  <b(u), 


This  function  is  (cf.  Art.  86) 


6)  =  e 


m=  +00 


where  Q  =  e 


THE   PROBLEM   OF   INVERSION.  163 


If  the  two  moduli  of  periodicity  A  and  B  have  the  property  that  the 

•efficient  of  i  in  ~—  is  positii 
B 

form  a  function  4>(w)  so  that 


coefficient  of  i  in  —  is  positive,  then  we  may  write  a  =  B  and  6  =  A  and 
B 


*(«)= 


m  =  +  oo  m2-  2  m 

—  — fr-  mu 


.  m=  —oo 


where  Qo  =  e"  Band  Bm+k=  Bm. 
We  then  have 


B}  = 


xik 

~~(*      A 


$>(u  +  A) 
Instead  of  the  variable  u  we  may  introduce  any  variable  quantity,  say 


u(z,  s)=  I       -• 

*SZQ,SO  S 

We  then  have 

$00  -^«(*,  «)]-¥(*,*),  say. 

It  is  seen  that  "^(z,  s)  is  a  function  of  position  in  the  Riemann  surface 
and  is  not  a  one-valued  function;  that  is,  when  z,  s  are  given,  ^(z,  s)  does 
not  take  one  definite  value.  For  u(z,  s)  depends  upon  the  path  of  inte 
gration,  so  that  (cf.  Art.  139) 

u(z,  s)  =  u(z,  s)  +  tnA  4-  nB. 

Hence  the  complex  of  values  "*&(z,  s)  which  belong  to  one  position  z,  s  is 
expressed  through 

\p(2>  s)   =  &[ti(z,  s)  +  mA  +  nB], 

where  m  and  n  are  integers. 

Since  4>(w)  has  the  period  B,  the  above  complex  of  values  reduces  to 

<b[u(z,  s)  +  mA]. 

We  saw  in  Art.  91  that  the  following  relation  existed  for  the  general 
^-function:  _  E**(2mM+W26) 


Consequently  the  complex  of  values  above  becomes 

-^[2mu(z,s)+m*A] 

<&[u(z,  s)  +  mA]  =  e  $["&  «)]• 

It  is  evident  that  &[u(z,  s)]  =  W(z,  s)  is  a  one-valued  function  of  position 


on  the  Riemann  surface   T'.     It  also  follows  that  between  "^(z,  s)  and 
¥(z,  s)  there  exists  the  relation 


The  integer  m  is  positive  or  negative  depending  upon  the  number  of  times 
the  path  of  integration  has  crossed  the  canal  a  and  upon  the  direction 
at  the  crossing. 


164  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  141.     We  saw  in  Art.  94  that 


Let  the  corresponding  "^-functions  be  denoted  by 


We  then  have,  for  example, 


= 


_xik 

It  follows  that 


- ~[2 mu(z,  s)  +m?A}  - 

B 


and  since  ^1(2,5),  ^^(z,  s)  are  both  one-valued  functions  of  position  on 
the  Riemann  surface,  it  is  also  seen  that  —  1    '     is  a  one-  valued  function 


of  position  on  the  Riemann  surface. 

The  functions  "^(z,  s)  =  $[u(z,  s)]  are  infinite  series  which  are  conver 
gent  for  all  values  of  the  argument  u(z,  s)  which  are  not  infinitely  large 
(Art.  86).  We  have  proved,  however,  that 


is  infinite  for  no  point  of  the  Riemann  surface,  including  the  point  at 
infinity.     It  follows  that  ^1(2,  s),  ^(z,  s)  are  everywhere  convergent  and 

\T/     f^     s\ 

consequently  the  quotient  —  1V  '    '  has  definite  values  everywhere  on  the 


Riemann  surface.  But  a  one-valued  function  of  z,  s  which  has  every 
where  a  definite  value  is  a  rational  function  of  z}  s.  It  follows  then  that 

)  _  R(      , 

where  R  denotes  a  rational  function. 

ART.  142.     Let  us  next  study  more  closely  some  of  the  subjects  which 
we  have  passed  over  rather  rapidly. 

We  had  on  the  canal  a  :  u(X)  —  u(p)  =  A, 
on  the  canal  b  :  u(p)  —  u(X)  =  B. 

It  made  no  difference  where  the  point  A,  p  was  situated  on  the  canal. 
We  may  therefore  take  the  point  a,  a'  where  the  canal  b  cuts  the  canal  a 
and  have  accordingly 

u(ar)  —  u(a)  =  A, 
or 

A  =   P  —  in  T',  (cf.  Neumann,  loc.  cit.,  p.  248], 

J  a     S 


THE   PROBLEM   OF   INVERSION; 


165 


the  integration  being  in  the  negative  direction.  In  the  T'-surface  we  may, 
starting  with  a,  follow  the  canal  b  around  to  the  point  a',  and  conse 
quently  have 

p. 


A-  f-i 

Jb   8 


the  integration  being  in  the 
negative  direction;  i.e.,  the 
quantity  .4  is  the  closed  in 
tegral  around  the  canal  b. 
In  the  same  way 

B  =  u(p)-u 


the  integration   being  in  the  negative  direction.     We  have  thus  shown 

'  that  B  is  the  closed  in- 

p  S"         tegral  over  the  canal  a. 

ART.  143.  In  the 
previous  discussions 
we  have  assumed  that 
R(z)  is  of  the  fourth 
degree  in  z.  When 
R(z)  is  of  the  third 
degree,  we  have  only 
three  finite  branch 
points,  a  i,  a2,  a3,  say. 
But  here  the  point  at  infinity  is  also  a  branch-point  (Art.  115).  We 
may  therefore  connect  a!  and  a2  by  a  canal  and  a3  with  the  point  at 
infinity.  The  Riemann  surface  may  then  be  represented  as  in  the 
former  case  (see  figure). 

A 

ART.  144.     In  the  derivation  of  the  function  4>(w)  the  ratio  —  cannot 

n 

be  real.  Following  the  methods  of  Riemann*  we  shall  show  that  this 
ratio  is  imaginary  and  that  the  coefficient  of  i  must  be  positive,  a  result 
which  was  also  necessary  in  the  previous  discussion. 

We  saw  that  u(z,  s)  was  a  one-valued  function  of  position  on  the  Riemann 
surface  T '.  All  functions  of  the  complex  variable  are  in  general  also 
complex,  and  we  may  consequently  write 

H(z,  s)  =  p  +  iq. 

*  Riemann,  Theorie  derAbeVschenFunctionen,  Crelle,  Bd.  54,  p.  145;  see  also  Koenigs- 
berger,  Elliptische  Functionen,  pp.  368,  369;  Fuchs,  Crelle,  Bd.  83,  pp.  13  et  seq. 


Fig.  62. 


166  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  quantity  u(z,  s)  is  everywhere  finite  in  T',  and  from  the  developments 
by  which  it  was  shown  always  to  be  finite,  it  is  readily  proved  to  be  also 
continuous. 

If  we  write  z  =  x  +  iy, 

then  p  and  q  are  everywhere  one-valued,  finite  and  continuous  functions 
of  x,  y. 

Noting  that     da(z>s)  -  du(z>  s)  **  =  du(z>  s)  .  1  =  **  =       1      . 
dx  dz        Ox  dz  dz       \/R(z) 

it  is  seen  that  —  is  infinite  for  z  =  01,  a2,  a3,  or  a4.     On  the  other  hand, 

du       dp   ,     .  00 

-    =   _£    _|_    l—Lj 

Ox       Ox         Ox 
and  consequently  either  -2  or  -^  or  both  of  these  derivatives  are  infinite 

for  z  =  ai,  a2,  a3,  or  a  4. 
Form  next  the  integral 


where  the  integration  is  to  be  taken  over  the  whole  boundary  of  the 
Riemann  surface  T'  .  This  surface,  see  figure  in  the  preceding  article, 
is  bounded  by  the  two  banks  \  and  p  of  the  two  canals  a  and  &.  It  is  seen 
that  we  may  go  over  both  the  banks  X  and  p  of  a  and  b  with  a  single  trace. 

The  integral  /  pdq  taken  over  this  trace  may  be  divided  into  several 
integrals  as  follows: 

//*&>)  ru)  /*u)  /*v» 

pdg  =   I       pdg  +   /       pd#  +   /         pdq  +   /       pdq, 
t/ar£on+a        •J09Jfon-»       t/^/a'on-a        t/a'^aon  +  ft    v 


where1  (^o)  as  an  upper  index  means  that  we  are  on  the  right  bank, 
means  the  portion  of  curve  gone  over,  and  +  a  means  on  the  canal  a  in 
the  positive  direction. 

ART.  145.     We  saw  above  that 

du  =  —^=  =  dp  +  idq, 
VR(z) 

dz^  =  dx^idy  =  d         .d 
VR(z)         VR(z) 

If  we  write  t  -   =  ^(x,  y)  +  ^(x,  y), 

VR(z) 
then  is 


(j)(x,  y)dx  —  ^r(x,  y)dy  +  i{  ^(x,  y)dx  +  <f>(x,  y)dy}  =  dp  4-  idq. 
It  follows  that  j/x 

W, 


THE   PROBLEM   OF   INVERSION.  167 

The  function  ^(x,  y),  which  is  the  coefficient  of  i  in  ,  will  have  at 


two  opposite  points  on  the  left  and  right  banks  of  the  canals  values  which 
are  different  only  by  an  infinitesimal  small  quantity,  since  the  canals  a 
and  b  are  indefinitely  narrow.  The  same  is  true  of  the  function  </>(x,  y). 
It  follows  that  dq  will  have  at  two  points  opposite  each  other  on  the 
canal  a  the  same  values,  but  the  signs  will  be  different,  since  the  integration 
at  these  points  has  been  taken  in  the  opposite  direction. 
We  may  therefore  write  the  above  integral  in  the  form 

/r     (p)     (A)  r     (P)     (A) 

Pdg  =  J+m*P  ~  P^dq  +  J+>{P  ~  P^dq- 
In  Art.  139  we  put  A  =  u(X)  —  u(p)  on  the  canal  a; 

(A)         (A)  (p)          (p) 

or  A  =  p  +  iq  —  {  p  +  iq  } 

(A)        (P)  (A)        (P) 

=  P  -  p  +  t\<i  -  q}- 

If  further  we  write  A  =  a  +  ifi,  then  is 

(A)        (P) 

a  =  p  —  p  on  the  canal  a. 
We  also  had 

(p)         (p)         (A)         (A) 

B  =  u(p)  —  Ti(X)  =  p  +  iq  —  {p  +  iq} 

(p)        (A)  (p)        (A) 

=  p-  P  +  i{q-q\, 

and  writing  B  =  r  +  id. 

(p)       (A) 

it  follows  that  7-  =  p  —  p  on  the  canal  b.     It  is  seen  at  once  that  the  above 
integral  may  be  written 


/  pdq  =  -  a  I  dq  +  r  I  dq. 

J  J+a  J+b 


+a  «/+* 

»  it  is  clear  thai 

VR(z) 


Since  du  =     ,  Z     >  it  is  clear  that 


-J$  - 


=  I  dp  +  i  I  dq. 

Jb  Jb 

Further,  since  A  =  a  +  if},  we  have 

/?  =  I  dq;  and  similarly 

*)  b 

8  =  fdq. 
<Ja 

The  integral  above  is  finally 

/  pdq  =  rfi  -  ad. 


168  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  146.  We  shall  calculate  the  same  integral  in  another  manner. 
Suppose  that  P  and  Q  are  real  functions  of  the  real  variables  x  and  y;  then  the 
curvilinear  integral 

+  Qdy), 


where  the  integration  is  taken  over  the  complete  boundary  of  a  region  within 
and  on  the  boundary  of  which  P  and  Q  together  with  their  partial  derivatives 
of  the  first  and  second  order  are  one-valued,  finite  and  continuous,  is  equal 
to  the  surface  integral 


. 

taken  over  the  same  region* 

Consider  the  curvilinear  integral 


where  as  above  the  integration  is  to  be  taken  over  both  banks  of  the 
two  canals  a  and  b  in  the  Riemann  surface  Tr.  We  have  seen  that  p  is 
one- valued,  finite  and  continuous  within  this  surface,  since  it  is  the  real 

part  of  v(z,  s).  But  (see  Art.  144)  -2-  and  -2-  become  infinite  at  the  points 
«i,  Oi2,  a3  and  a4. 

Hence  to  apply  the  theorem  just  stated,  we  must  cut  these  points  out 
of  the  surface  by  means  of  very  small  double  circles.  The  resulting 
Riemann  surface  call  T".  In  this  surface  the  conditions  required  are 
satisfied.  The  curvilinear  integral  must  now  also  be  taken  over  the 
double  circles.  But  as  shown  in  Art.  137  the  integrals  over  these  double 
circles  are  zero. 

If  then  we  write  in  the  formula 


instead  of  Pdx  +  Qdy  the  quantity  p  —  dx  +  p-^dy, 
we  will  have  to  substitute  y 

c  ao  ,  i  •      ap  do         d^a 

for  —  the  expression  -£-  -*  +  p  — *- 

ax  ax  ay        axay 

T  <«  dP  ,1  •        dv  da  d2q 

and  for  —  the  expression  -*-  -^-  +  p  — *-: 

ay  ay  ax         axay 

and  consequently 

//^  Crdp   do       dp  do~\~ 
paq  =     •       \  -A-  — * —  — *-  axoy. 
J  J  |_ax  ay     ay  axj 

*  Forsyth,  p.  23;  see  also  Casorati,  Teorica  delle  funzioni  di  varidbili  complesse, 
pp.  64-69;  Neumann,  AbeVsche  Integrate,  2d  ed.,  p.  390.  Schwarz,  Ges.  Werke,  Bd.  II, 
has  shown  that  there  are  certain  limitations  of  this  theorem;  and  Picard,  Traite  d" Analyse, 
t.  2,  pp.  38  et  seq. 


THE   PROBLEM   OF   INVERSION.  169 

But  since  **-  =  &  and  &  =  -  & 

Ox     dy        ay        dx 

(being  the  conditions  that  u(z,  s)=  p  +  iq  have  a  definite  derivative), 
and  since  I  pdg  =  (5/-  —  ad,  it  follows  that 


As  the  elements  under  the  sign  of  integration  are  essentially  positive, 
it  is  seen  that  fa  —  ad  is  a  positive  quantity. 
But  we  have 

B  =  r  +  i^  =  (r  +  id  )  (a  -  J3)  =  ar  +  pd    ,    .  ad  -  fa 
A       a+ip  a2  +  ^  ~~   a2  +  p2  ~    ?   a2  +  p2  ' 

D 

Since  ad  —  fa  is  different  from  zero,  the  ratio  —is  not    real,*  and   the 

B  A 

coefficient  of  i  in  —is  negative;  hence  the  coefficient  of  i  in  ^-is  positive. 
A  B 

We  may  therefore  (see  Art.  86)  form  functions  $(u)  such  that 
*(«  +  B)  -$(*), 


•01 

"      a     A 


ART.  147.     In  the  expression 


since  w(2,  s)  is  always  finite,  the  exponential  factor  is  always  finite  so  long 
as  m  is  finite.  Further,  since  &  is  only  infinite  for  infinite  values  of  its 
argument,  it  follows  that 

¥(*,*)  =$[u(z,s)] 

is  never  infinite.     Hence  also  ¥(z,  s)  is  only  infinite  when  m  is  infinite. 
It  is  also  evident  that  ^(z,  s)  can  only  be  zero  when  ¥(z,  s)  —  0. 
We  shall  now  see  how  often  the  function  W(z,  s)  becomes  zero  on  the 
Riemann  surface  Tr. 

In  Art.  92  we  saw  that  if  a  f  unction  f(z,  s)  is  discontinuous  at  isolated 
positions  within  a  portion  of  surface,  but  otherwise  is  one-valued  and 
finite,  then 

log/fcs)  . 
dz 

where  the  integration  is  taken  over  the  complete  boundary  of  the  portion 
of  surface,  is  equal  to  the  sum  of  the  orders  of  the  zeros  of  the  function 

*  Cf.  Thomae,  Abriss  einer  Theorie  der  Functionen,  etc.,  p.  102;  Falk,  Ada  Math., 
Bd.  70;  Pringsheim,  Math.  Ann.,  Bd.  27. 


170  THEORY   OF   ELLIPTIC   FUNCTIONS. 

diminished  by  the  sum  of  the  orders  of  its  infinities  within  the  portion  of 
surface  in  question;  i.e., 


\mj  ' 


2m  J          dz 

As  the  portion  of  surface  we  shall  take  the  surface  T'  which  is  bounded 
by  the  canals  a  and  fe,  and  for/(z,  s)  we  have  here  ^(2,  s).  There  being 
no  infinities,  HA  =  0,  and  consequently 


j__  rd 

2  Tti  J 


,  s)  d 


dz 

where  the  integration  taken  over  both  banks  of  the  canals  a  and  b  is  equal 
to  the  sum  of  the  orders  of  the  zeros  in  T' ' . 

Now  on  the  canal  a  we  have    u(X)—  u(p)=  A, 

or  u(X)=  u(p)  +  A. 

It  follows  that 


=  <S>[u(p)  +  A]  = 
and  consequently  that 


On  the  canal  b  we  have       u(p)—  u(X}=  B, 
or  u(X)  =  u(p)  —  B. 

It  follows  that 


or 


From  the  figure  in  Art.  142  it  is  seen  that 

r  d\oKv(z,8)  dz  =  w  d  log  v(Z,  s)  dz  +  p»  d\ 

Jr  dz  Jarfon+a       dz  Jpdpon-6 

+   /•(«     d  log  V(z,  s)  dz  +    rv    d  log  V(z,  s) 

Jy/a'on-a   ^Z  Ja' 

I   -/  aT^ 

J+adz[       ^(^ 


'd'aon+b 


which  owing  to  (M)  and  (N) 


dz 


THE   PROBLEM   OF   INVERSION.  171 

But  from  Art.  139  it  is  seen  that 


We  therefore  have  finally 


i  pnoE 

!  -i  J 


dz 


It  is  thus  seen  that  the  intermediary  function  W(z,  s)  has  k  zeros  on  the 
surface  T';  and  since  W(z,  s)  vanishes  on  the  same  points  as  ^(z,  s),  it 
follows  that  W(z,  s)  has  k  zeros  on  the  Riemann  surface  T. 

ART.  148.     We  saw  (Art.  87)  that  when  k  =  2 


Further,  write  Q  =  q*,  and  it  follows  that 


If  in  ®i(u)  we  write  —  «  in  the  place  of  //,  the  summation  is  not  thereby 
changed,  and  we  have 


/(=-» 


From  this  it  is  seen  that  ®i(u)  =  ®i(—u),  or  ®i(w)  is  an  even  function. 
Similarly  writing  —  «  —  1  for  ,«  in  the  formula  for  HI(M)  we  have 


or  HI(M)  =  HI(  —  u),  so  that  this  function  is  also  even. 

ART.  149.     If  in  ©I(M)  we  write  u(z,  s)  instead  of  u,  then  ®I(M)  becomes 


¥<,(*,  s)  =  %(z,  s)  .  e     B  (cf.  Art.  140). 

Suppose  that,  starting  from  a  point  z0,  SQ  in  the  upper  leaf  of  the  Riemann 
surface  T'  ',  a  path  of  integration  is  taken  to  the  point  z,  s,  which  may  cross 
the  canals  a  and  b  as  often  as  we  choose.  The  point  z,  s  may  lie  in  either 
the  upper  or  the  lower  leaf.  Next  starting  from  the  point  z0,  —  s0,  which 
lies  immediately  under  the  point  z0,  SQ,  let  us  construct  a  second  path,  which 
is  everywhere  congruent  to  the  first  path,  that  is,  which  lies  in  the  under 


172  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

leaf  when  the  first  path  is  in  the  upper,  and  is  in  the  upper  leaf  when  the 
first  path  is  in  the  under.  If  further  we  form  the  integral  of  the  first  kind 
u(zy  s)  for  each  of  these  two  paths,  and  add  the  two  integrals,  it  is  seen 
that  the  elements  of  integration  cancel  in  pairs,  so  that 


where  (I)  and  (II)  are  used  to  denote  the  paths  of  integration.  Suppose 
that  20,  SQ  coincides  with  one  of  the  branch-points,  for  example  with  01, 
then  ZQ,  SQ  and  ZQ,  —  s0  coincide,  and  we  have 

/  (*z>s  dz         rz'~s  rlz 

I     S£+  I        *  =  o, 

«/ai        S         t/ai  S 

(I)  (II) 


or 


u(z,  s)  +  gA  +  hB  +  u(z,  -  s)  +  g' A  +  h'B  =  0, 


where  g,  g',  h,  hr  denote  integers. 
It  follows  that 

u(z,  s)  +  u(z,  -  s)  =  TA  +  dB, 

where  f  and  d  are  integers. 

//  then  we  take  a  branch-point  as  the  initial  point  of  the  path  of  integration, 
the  function  u(z,  s)  has  at  two  points  situated  the  one  over  the  other  in  the 
Riemann  surface  Tf,  values  whose  sum  is  equal  to  integral  multiples  of 
A  and  B. 

ART.  150.  If  we  write  u(z,  s)  for  u  in  ®i(u),  we  have  the  function 
W0(z,  s);  similarly  let  M*i(z,  s)  denote  the  result  of  substituting  u(z^s)  for 
u  in  Hi(V).  Then  noting  the  relations  existing  between  M*0,  "^o  and 
between  M^  and  "^i,  it  is  seen  (cf.  Art.  141)  that 


where  R(z,  s)  denotes  a  rational  function  of  its  arguments. 
It  will  be  shown  in  the  following  Chapter  that 

R(z,s)  =  g(z)  +  s  -h(z), 

where  g(z)  and  h(z)  are  rational  functions  of  z  alone. 
We  form  next  ^0(z,  -  s)  =  &i[u(z,  -  s)] 

'S)      *!&.-»)       Ktu^-s)] 
6i[-  tZ(g,g)  +  rA  +  dB]  _  0i[-  u(z,s)] 


HI[-  u(z,  s)  +  rA 

as  is  seen  from  the  functional  equations  which  ®i  and  HI  satisfy.     Since 
®i  and  HI  are  even  functions,  it  follows  that 

R(z,  -  »)  -  -  *(*'  s)' 


THE   PROBLEM   OF   INVERSION.  173 

We  therefore  have 

g(z)  -  s-h(z)  =  g(z)  +  s-h(z), 
and  consequently 

s  -h(z)  =  0. 

Since  s  is  not  identically  zero,  we  must  have 

h(z)  =  0; 


or  R(z,  s)  is  a  rational  function  of  z  alone. 

ART.  151.  Since  A:  =  2,  it  follows  that  HI  and  ©i  have  two  zeros  of 
the  first  order  on  the  Riemann  surface;  and  since  the  quotient  of  these 
two  functions  is  a  rational  function  of  z  it  is  evident  that 

(M) 


i       A3z  +  A4 

where  the  A's  are  constants.     This  function  has  the  two  zeros  of  the  first 
order 

and  the  two  infinities 


Remark.  — If  the  zero  z  =  -  ^  is  a  branch-point,  say  ai,  then  (see 

1  A 

Art.  120)  twice  the  exponent  of  the  lowest  power  of  2  —  ai  =  z  +  —  in 

A  i 
the    development  in  ascending   powers  of  z  —  a-i    is   the    order  of  the 

zero.     But  as  the  development  of  the  numerator  of  the  above  expression 

is   simply   AI  \z  +  ^  I  it   is  seen  that  2  is    the  order  of  the   zero  for 

4o         L  ^ 

z  =  -  ^  •     Such  a  zero  is  therefore  to  be  counted  as  two  zeros  of  the  first 

1  4. 

order.      The  case  where  —  ^p  is  a  branch-point    may  be  treated   in  an 

analogous  manner. 

ART.  152.     It  follows  directly  from  equation  (M)  above  that 


from  which  it  is  seen  that  z  is  a  one-valued  doubly  periodic  function  of  u 
with  periods  A  and  B.      We  call  z  the  inverse  of  the  elliptic  integral  u,  where 

dz 


-  r 

J 


-  a<2}(z  -  a3}(z  —  a4) 

Although  u  is  not  a  one-valued  function  of  z  (Art.  139),  the  inverse 
function  z  is  one-valued  in  u.  The  constant  A  under  the  radical  is  of 
course  not  the  same  constant  as  the  period  A . 


174  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  may  also  note  that  ^ 

s  =  — — 
du 

is  a  one-valued  function  of  u;  for  the  derivative  of  a  one-valued  doubly  peri 
odic  function  is  one-valued  and  doubly  periodic. 

ART.  153.  The  following  remarks  of  Lejeune  Dirichlet  (Geddchtniss- 
rede  auf  Jacobi;  Jacobi's  Werke,  Bd.  I,  pp.  9  and  10)  are  instructive  and 
historical: 

"  Es  ist  Legendres  unverganglicher  Ruhm  in  den  eben  erwahnten 
Entdeckungen  die  Keime  eines  wichtigen  Zweiges  der  Analysis  erkannt 
und  durch  die  Arbeit  eines  halben  Lebens  auf  diesen  Grundlagen  eine 
selbstandige  Theorie  errichtet  zu  haben,  welche  alle  Integrale  umfasst, 
in  denen  keine  andere  Irrationalitat  enthalten  ist  als  eine  Quadratwurzel, 
unter  welcher  die  Veranderliche  den  4ten  Grad  merit  iibersteigt.  Schon 
Euler  hatte  bemerkt,  mit  welchen  Modificationen  sein  Satz  auf  solche 
Integrale  ausgedehnt  werden  kann;  Legendre,  indem  er  von  dem  gliick- 
lichen  Gedanken  ausging,  alle  diese  Integrale  auf  feste  canonische  Formen 
zuriickzufiihren,  gelangte  zu  der  fur  die  Ausbildung  der  Theorie  so  wichtig 
gewordenen  Erkenntniss,  dass  sie  in  drei  wesentlich  verschiedene  Gat- 
tungen  zerfallen.  Indem  er  dann  jede  Gattung  einer  sorgfaltigen  Unter- 
suchung  unterwarf,  entdeckte  er  viele  ihrer  wichtigsten  Eigenschaften, 
von  welchen  namentlich  die,  welche  der  dritten  Gattung  zukommen, 
sehr  verborgen  und  umgemein  schwer  zuganglich  waren.  Nur  durch  die 
ausdaurerndste  Beharrlichkeit,  die  den  grossen  Mathematiker  immer  von 
neuem  auf  den  Gegenstand  zuriickkommen  liess,  gelang  es  ihm  hier 
Schwierigkeiten  zu  besiegen,  welche  mit  den  Hiilfsmitteln,  die  ihm  zu 
Gebote  standen,  kaum  iiberwindlich  sheinen  mussten.  .  .  . 

"  Wahrend  die  friiheren  Bearbeiter  dieses  Gegenstandes  das  elliptische 
Integral  der  ersten  Gattung  als  eine  Function  seiner  Grenze  ansahen, 
erkannten  Abel  und  Jacobi  unabhangig  von  einander,  wenn  auch  der 
erstere  einige  Monate  friiher,  die  Nothwendigkeit  die  Betrachtungsweise 
umzukehren  und  die  Grenze  nebst  zwei  einfachen  von  ihr  abhangigen 
Grossen,  die  so  unzertrennlich  mit  ihr  verbunden  sind  wie  der  Sinus 
zum  Cosinus  gehort,  als  Functionen  des  Integrals  zu  behandeln,  gerade 
wie  man  schon  friiher  zur  Erkenntniss  der  wichtigsten  Eigenschaften  der 
vom  Kreise  abhangigen  Transcendenten  gelangt  war,  indem  man  den 
Sinus  und  Cosinus  als  Functionen  des  Bogens  und  nicht  diesen  als  eine 
Function  von  jenen  betrachtete. 

"  Ein  zweiter  Abel  und  Jacobi  gemeinsamer  Gedanke,  der  Gedanke 
das  Imaginare  in  diese  Theorie  einzufiihren,  war  von  noch  grosserer 
Bedeutung  und  Jacobi  hat  es  spater  oft  wiederholt,  dass  die  Ein- 
fiihrung  des  Imaginaren  allein  alle  Rathsel  der  friiheren  Theorie  gelost 
habe." 


THE   PROBLEM   OF    INVERSION.  175 

ART.  154.  If  we  had  not  wished  to  study  the  one-valued  functions  of 
position  on  the  Riemann  surface  s  =  \/R(z)t  we  might  have  shown 
immediately  that  //7,\2 

©-««• 

For  in  the  differential  equation  (cf.  Art.  106) 

'  -^ 

when  a  definite  value  is  given  to  z,  say  z0,  then  the  sum  of  the  two  roots 

of  the  equation  is  ^  v        /^x  ^ 

I   -j- 1       \  = 


\dz/2         A0(z0)° 

On  the  other  hand,  corresponding  to  the  value  z0  there  are  within  the 
initial  period-parallelogram  two  values  of  u  say  ui  and  u2.  Also,  since 
u\  +  u2  =  Constant,  it  follows  that 

!*!•.- A  (ii) 

But  the  left-hand  side  of  (i)  is  the  same  as  the  left-hand  side  of  (ii),  and 
consequently*  A  1(2)  =  0. 

ART.  155.  A  Theorem  due  to  Liouville.  Suppose  that  w  =  F(u)  is 
a  doubly  periodic  function  of  the  fcth  order  with  periods  a  and  6;  also  let 
2  =  /(«)  be  a  doubly  periodic  function  of  the  second  order  with  the  same 
periods.  There  exists  then  (see  Art.  104)  an  integral  algebraic  equation 

of  the  form  G(w,  z)  =  0, 

which  is  of  the  second  degree  in  w  and  of  the  fcth  degree  in  z. 
This  equation  may  be  written 

Lw2  +  2Pw  +  Q  =  0, 

L,  P  and  Q  being  integral  functions  of  degree  not  greater  than  k  in  z. 

It  follows  that  

-  P±\  P2-LQ        -P+  a 

L  L~ 

where  a  =  ±VP2  -  LQ. 

We  therefore  have  7-       ,    p 

O  —  Ljii    ~r  -t   j 

so  that  o  is  a  one- valued  function  of  w. 

We  saw  above  that  ^  

—  =  ±\/R(z). 
du 

*  Cf.  Harkness  and  Morley,  Theory  of  Functions,  p.  293,  where  numerous  other 
references  are  given. 


176  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  is  also  seen  that  corresponding  to  one  value  of  z  there  are  two  values 
of  a  differing  only  in  sign,  and  corresponding  to  this  same  value  of  z  there 

are  two  values  of  — which  differ  only  in  sign. 
du 

Hence  T(z)  =  a  -±-(dz/du)  is  a  one-valued  function  of  z  with  periods 
a  and  b.  It  follows  also  (see  Art.  104)  that  an  algebraic  equation 
exists  between  a  -r-(dz/du)  and  z;  and  consequently  a  -±-(dz/du)  is  inde 
terminate  for  no  value  of  u.  But  a  one-valued  function  which  has  no 
essential  singularity  is  a  rational  function  (Chapter  I).  Hence  T(z)  is  a 
rational  function  of  z. 

It  is  also  seen  that  * 

-  P  +  T(z)  — 

— ^  =  p  +  qs, 

p  and  q  being  rational  functions  of  z. 

We  have  thus  shown*  that  w  may  be  expressed  rationally  in  z  and  s  =  — ; 

du 

or  w  =  R(z,  s),  which  theorem  is  due  to  Liouville. 

ART.  156.  A  Theorem  of  Briot  and  Bouquet  (Fonctions  Elliptiques, 
p.  278).  Suppose  that  w  =  F(u)  is  a  doubly  periodic  function  of  the 
&th  order  with  primitive  periods  a  and  b  and  let  t  =  fi  (u)  denote  any 
other  doubly  periodic  function  with  the  same  periods.  We  shall  show 

that  t  is  a  rational  function  of  w  and  —  • 

du  , 

There  exists  (Art.   104)  between   w  and  w'  =  —  an  integral  algebraic 

du 

equation 

(I)  G(w,  w')  =  0, 

which  is  of  the  Mh  degree  in  w'. 

Hence  corresponding  to  one  value  of  w  there  correspond  in  general 
k  values  of  w'  in  a  period-parallelogram.  Suppose  that  for  the  value 
WQ  there  correspond  the  k  values 

Wi',W2',     •     •     •    ,    Wk'.  (1) 

• 

Further,  since  w  is  of  the  /bth  order,  there  correspond  k  values  of  u  to  WQ  in 
the  period-parallelogram,  say 

HI,  u2,  ..;,***.  (2) 

We  also  know  that  between  the  functions  /  and  w  there  is  an  algebraic 
equation 

(II)  (?!  (w,0=0 

of  the  fcth  degree  in  tt  so  that  corresponding  to  the  value  w0  there  are 
k  values  of  t,  say  ^  t2,  .  .  .  ,  tk.  (3) 

*  Liouville,  Crelle,  Bd.  88,  p.  277,  and  Comptes  Rendus,  t.  32,  p.  450. 


THE   PROBLEM   OF   INVERSION.  177 

We  note  that  the  system  of  values  (3)  correspond  to  the  system  of  values 
(1)  in  such  a  way  that  to  every  system  of  values  (w,  w')  there  corresponds 
one  definite  value  of  t  and  only  one. 
The  functions 

tw',tw'2,  .  .  .  ,  tw'k~l 

enjoy  the  same  property. 
It  follows  that  the  sums 

/i  +  t2  +t3          +  •   •   -  +  tk         =  PQ, 

+  t2w2'       +  t3w3'     +  •   •   •  +  tkwk'    =  PI, 

+  t2W2'2       +  t3W3'2    +   •     •     •   +  tkWk'2    -  P2, 

fiwi'*-1  +  t2w2'k~l  +  faM'a'*-1  +  •   •   -  +  to'*-1  =  P*-i 

are  one-  valued  functions  of  w,  and  have  definite  values  for  all  values  of  w 
on  the  Riemann  surface.     They  are  therefore  rational  functions  of  w. 
ART.  157.     If  we  multiply  the  above  equations  respectively  by 

Ak-i,  Ak-2,  •   •  •  ^i,l, 
add  the  results  and  equate  to  zero  the  coefficients  of 

t-2,  £3,  •  •  •  >  tkj 
we  will  have  the  system  of  k  equations: 

~*  +  •-.-  +  Ak-2W2'  +  Ak-i  =  0, 
~*  +  •  •  •  +  Ak-2wB'  +  Ak-i  =  0, 


(4) 
and  the  additional  equation 


=  PA-I  +  A'iPjt-2  +  -  •  •  +  --U-2Pi  +  Ak-iPo.  (5) 

The  equations  (4)  show  that  the  quantities  ^.i,  A2,  .  .  .  ,  Ak-i  are  the 
coefficients  of  an  algebraic  equation  of  the  ft—  1st  degree  whose  roots  are 
w2',  w3',  .  .  .  ,  wkf. 

We  obtain  this  equation  by  dividing  (I)  arranged  in  decreasing  powers 
of  wr  by  w'  —  -M'I'.  The  coefficients  of  the  quotient,  which  are  integral 
functions  of  w0  and  wi',  will  give  the  quantities  .-ii,  A2,  .  .  .  ,  Ak-i- 

From  equation  (5)  we  have  t  expressed  as  a  rational  function  of  w  and  w'. 

This  theorem  is  a  generalization  of  Liouville's  Theorem  above.  In 
Chapter  XX  we  shall  again  prove  indirectly  both  theorems. 


178  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  158.  We  shall  prove  in  Chapter  XVI  that  the  doubly  periodic 
function  of  the  second  order  z  =  (j>(u)  is  such  that  <j>(u  +  v)  may  be 
expressed  rationally  in  terms  of  </>(u),  (/>'(u),  <f>(v)t  (£>'(v),  say 

<l>(u  +  v)=  Ri[<j>(u),  <l>'(u),  </>  (v),  P(v)],  (1) 

where  R  with  a  suffix  denotes  a  rational  function,  and  consequently  also 

f(u  +  v)  =  Rd&(u),  <j>'(u),  <f>(v),  $  (v)].  (2) 

For  the  present  admit  the  above  statements. 

By  Liouville's  Theorem  it  follows  that  w  =  F(u)  is  a  rational  function 
of  4,(u)  and  p(u),  or       p(u}  _  BJ[^(M))  ^(M)]. 
We  consequently  have 

F(u  +  v)  =  R3[<f}(u  +  v),  <j>'(u  +  v)] 

=  R^(u),<t>'(u},<j>(v),<i>'(v)}.  (3) 

Also  from  Briot  and  Bouquet's  Theorem 


and  <j>'(u)=R6[F(u),F'(u)]. 

Hence  from  (3)  we  see  that 

F(u  +  v)=  R7[F(u),  F'(u),  F(v),  F'(v)]. 

It  has  therefore  been  proved,  since  w  satisfies  the  latent  test  expressed 
by  the  eliminant  equation,  that  this  function  has  an  algebraic  addition- 
theorem,  and  in  fact  is  such  *  that  F(u  +  v)  may  be  expressed  rationally 
in  terms  ofF(u),  F'(u),  F(v),  F'(v). 

This  property,  see  Chapter  II,  also  belongs  to  the  rational  functions 
and  to  the  simply  periodic  functions. 

It  has  thus  been  demonstrated  that  to  any  one-valued  function  <j)(u) 
which  has  everywhere  in  the  finite  portion  of  the  plane  the  character  of  an 
integral  or  (fractional}  rational  function,  belongs  the  property  that  <f>(u  +  v) 
is  rationally  expressible  through  <f>(u),  (£>'(u),  <t>(v),  (/>'(v).  As  it  was  shown 
in  Art.  74  that  a  one-valued  analytic  function  cannot  have  more  than 
two  periods,  it  follows  (cf.  also  Art.  41)  that  a  one-  valued  analytic  function 
which  has  an  algebraic  addition-theorem  is  either 

I,  a  rational  function  of  u,   niu 

II,  a  rational  function  of  e  "  , 

III,  a  rational  function  of  z  and  — 

du 

The  first  two  cases  (Art.  41)  are  limiting  cases  of  the  third.  Every  tran 
scendental  one-valued  analytic  function  which  has  an  algebraic  addition- 
theorem  is  necessarily  a  simply  or  a  doubly  periodic  function. 

*  See  Schwarz,  Ges.  Math.  AbhandL,  Vol.  II,  p.  265. 


THE   PROBLEM   OF   INVERSION.  179 

ART.  159.  We  have  seen  that  any  rational  function  of  z  and  s  is  a  one- 
valued  function  of  position  on  the  Riemann  surface  s.  Hence  the  function 
w  of  the  preceding  article,  which  is  the  most  general  one-valued  doubly 
periodic  function,  is  a  one-valued  function  of  position  on  the  Riemann  sur 
face.*  The  quantity  s  is  the  root  of  the  algebraic  equation 

s2  -  R(z)  =  0, 

and  by  adjoining  this  algebraic  quantity  to  the  realm  of  rational  quanti 
ties  (z)  we  have  the  more  extended  realm  (z,  s)  composed  of  all  rational 
functions  of  both  z  and  s.  This  latter  realm  includes  the  former.  Since 
all  functions  of  the  realm  (z,  s)  are  one-valued  functions  of  position  on 
the  Riemann  surface  T  and  since  this  surface  is  of  deficiency  or  order 
unity,  we  may  say  the  realm  (s,  2),  the  elliptic  realm,  is  of  the  first  order, 
the  realm  of  rational  functions  (z)  being  of  the  zero  order. 

We  thus  see  that  the  study  of  functions  belonging  to  the  realm  of  order 
unity  is  coincident  with  the  study  of  the  doubly  periodic  functions  and  in 
fact  the  study  of  one  necessitates  the  study  of  the  other. 

The  elliptic  or  doubly  periodic  realm  (s,  2),  where 


s  =  V'A  (z  -  ai)  (z  -  a2)  (z  -  a3)  (z  -  a4)  =  ~, 

du 

degenerates  into  the  simply  periodic  realm  when  any  pair  of  branch 
points  are  equal  and  into  the  realm  of  rational  functions  (z)  when  two 
pairs  of  branch-points  are  equal  (including  of  course  the  case  where  all 
the  branch-points  are  equal). 

Thus  the  elliptic  realm  (z,  s)  includes  the  three  classes  of  one-valued 
functions  : 

First,  the  rational  functions, 

Second,  the  simply  periodic  functions, 

Third,  the  doubly  periodic  functions. 

All  these  functions,  and  only  these,  have  algebraic  addition-theorems. 
In  other  words,  all  functions  of  the  realm  (z,  s)  have  algebraic  addition- 
theorems,  and  no  one-valued  function  that  does  not  belong  to  this  realm  has 
an  algebraic  addition-theorem.  We  have  thus  proved  that  the  one-valued 
functions  of  position  on  the  Riemann  surface 

s2  =  R(z), 

R  denoting  an  integral  function  of  the  third  or  fourth  degree  in  z,  belong  to 
the  closed  realm  (z,  s)  of  order  unity,  and  all  elements  of  this  realm  and  no 
others  have  algebraic  addition-theorems. 

*  Cf.  Klein,  Theorie  der  elliptischen  Modulfunctionen,  Bd.  I,  pp.  147  and  539. 


CHAPTER   VIII 
ELLIPTIC  INTEGRALS  IN  GENERAL 

The  three  kinds  of  elliptic  integrals.     Normal  forms. 

ARTICLE  160.  At  the  end  of  the  last  Chapter  we  saw  that  the  most 
general  elliptic  function  could  be  expressed  as  a  rational  function  of  z,  s. 
We  shall  now  consider  the  integral  of  such  an  expression.* 

Let  RI(Z,  s)  denote  a  rational  function  of  z,  s.  This  function  may  be 
written  in  the  form 

R  (z  s)  =  A°  +  AlS  +  A*s2  +  '   '   '  +  Aksk 
B0  +  BlS  +  B2s2  +  •   •   -  +  Bis1 

where  the  A's  and  B's  are  integral  functions  of  z.  Owing  to  the  relation 
s2  =  A(z  —  a\)  (z  —  a2)  (z  —  a3)  (z  —  a4),  it  is  seen  that  the  even  powers 
of  s  are  integral  functions  of  z,  while  the  odd  powers  of  s  are  equal  to  an 
integral  function  of  z  multiplied  by  s,  so  that 


_C±Ds 
~~E~ 
where  C,  D,  and  E  are  integral  functions  of  z  as  are  A0',  AI,  BQ'  and 

Writing  |=  p(z)  and  5=  q(g)j 

EJ  E/ 

it  is  seen  that        D  /      \         /  \         /  \  /  \ 

Ri(z,  s)  =  p(z)  +  q(z)*s  =  p(z) 

s 

where  q(z)'S2  =  Q(z)  and  where  p(z),  q(z),  and  Q(z)  are  rational  functions 
of  z.     (See  also  Arts.  125  et  seq.) 
Consider  next  the  integral 


C 


,  s)dz  = 


The  first  integral  on  the  right  may  be  reduced  at  once  to  elementary 
integrals,  so  that  we  may  confine  our  attention  to  the  integral 

JSM  dz  which  may  be  written  /    -^     dz. 
«  J  VZ 


f(z)  denoting  a  rational  function  of  z,  and  s  =\/~R(z). 

*  Legendre,  Memoire  sur  les  transcendantes  elliptiques,  1794.     See  also  Legendre, 
Fonctions  Elliptiques,  t.  I,  Chap.  I. 

180 


ELLIPTIC    INTEGRALS   IN    GENERAL.  181 

ART.  161.     Suppose*  in  general  that 

R(z)  =  C0zn  +  C1z"-1  +  .  •   • -f  Cn, 
where  the  C's  are  constants.     When  n  is  greater  than  4,  the  integral 


dz 


is  no  longer  an  elliptic  but  a  hyper  elliptic  integral;  when  n  =  3  or  4  we 
have  the  elliptic  integrals,  and  when  n  =  2  we  have  the  integrals  that 
are  connected  with  the  circular  functions. 
The  rational  function  f(z)  may  be  written 

f(z)  =  ^  =  G(z)  +  ^I&, 
g(z)  g(z) 

the  g'a  and  G's  denoting  integral  functions,  and  say 

g(z)  =  B(z  -  b^  (z  -  b*)**(z  -  b^  .   .   •  . 
Hence  when  resolved  into  partial  fractions 

/(z)  =  G(z)  +2) — ^—>>  (Ak  constants), 

i    (z  -  bi)** 
and  also 

dz 


Since  G(z)  is  an  integral  function,  the  first  integral  on  the  right-hand  side 
may  be  resolved  into  a  number  of  integrals  of  the  form 

ink          •' 

\/R(z} 

We  thus  have  two  general  types  of  integrals  to  consider, 


and  Hk=  C 

J  (z- 


VR(z) 
dz 


(z-b)><VR(z) 
ART.  162.     Form  the  expression 


=  kzk~l\/R(z)  -   .=zk  =  _[<2kR(z)  +  zR'(z)] 

%VR(z}  2VR(z) 


- 
2VR(z) 

iZ  +  2kCn]. 


*  Briot  et  Bouquet,  Fonctwns  EUiptiques,  p.  436;  see  also  Koenigsberger,  EUip- 
tische  Functionen,  p.  260;  Appell  et  Lacour,  Fonctions  EUiptiques,  p.  235. 


182  THEORY  OF   ELLIPTIC   FUNCTIONS. 

It  follows  through  integration  that 

2zkVR(z}  =  (2k  +  n)C0Ik  +  n-i  +  (2k  +  n-  l)Cl!k+n-2 

+  (2k 


If  in  this  expression  we  put  A;  =  0,  it  is  seen  that  In-i  may  be  expressed 
through  In-2,  In-s,  •  •  -  ,  I Q,  1  - 1  and  through  the  function  VR(z);  when 
k  is  put  =  1,  we  mayjexpress  In  through  In- 1,  In-2,  •  •  •  ,  /o  and  through 
the  function  z\/R(z).  If  further  we  write  for  In-\  its  value,  we  may 
express  In  through  In-2,  In-s,  ..  .  •  ,  /Q,  J-i  and  an  algebraic  function. 
This  algebraic  function  is  an  integral  function  of  the  first  degree  in  z 
multiplied  by  \/R(z). 

Continuing  in  the  same  manner,  we  may  express  In  +  X  through  In-2, 
In-3,  -  •  •  ,  IQ,  I  - 1  and  an  algebraic  function  which  is  an  integral  function 
of  the  ^  +  1  degree  in  z  multiplied  by  \/R(z). 

ART.  163.  We  consider  next  the  integrals  of  the  type  Hk.  Form  the 
expression 

^  ["  VR(z)  1  ^ k         v^Tz)  +  -          R'^ 

dzL(z-b)k\  (z-b)k+1  2  (z-b)kVR^zj 

-R'(z)(z-b)}. 


/       - 
2VR(z)(z- 

If  we  write  -  2kR(z)  +  R'(z)  (z-b)= 

then  is 

z)  +  R(v  +  V(z)(z  -  6) 


or  <l>W(z)  =  (v  -  2k)RW(z)  +  (z  - 

It  follows,  since 


that  21 

=-  2kR(b) 


(n  —  1)1 


(z  - 


n  ~  l  ~^  k 
(n  —  1)! 


ELLIPTIC   INTEGRALS   IN   GENERAL.  183 

Integrating  it  is  seen  that 


(n-V  (6)  Hk-n+2 


(n  -  1)! 
If  we  put  k  =  1,  we  see  that  H2  ma\r  be  expressed  through 


This  is  correct  only  if  R(b)j£  0;  i.e.,  if  6  is  not  a  root  of  the  equation 
R(z)=  0.     This  case  is  for  the  moment  excluded.     We  note  that 

rr         C    dz          1  .  „  f(z  -  6)  dz       r        7  r  . 

HO  =  I  —  =====  =  70,  H-i  =   I  -        -  •  •  • 

J  VRz  J     \ 


R(z) 
„  C 

n.  -(n-2)=     I 

J 


VR(z) 

From  this  it  is  seen  that  the  integrals  HQj  H-i,  H  -2,  .  .  .  ,  #_(n_2)  may 
be  expressed  through  integrals  of  the  type  /A-.  Hence  the  integral  HI 
alone  offers  something  new. 

We  note  that  H2  may  be  expressed  through  H\,  IQ,  I\,  .  .  .  ,  7n-2  and 
through  an  algebraic  function  of  z.     If  we  put  k  =  2,  we  may  express  H3 


« 

through  H2,  HI,  .  .  .  ,  #_(n_3)  and  through  -  -J-^  ;  or,  if  for  H2  we  write 

(z-by 

its  value  just  found.  H3  may  be  expressed  through  //1?  70,  /i,  .  .  .  ,  In-2 
and  an  algebraic  function  of  z.  In  general,  we  may  express  Hm  through 
HI,  I0,  1  1,  .  .  .  ,  7,j_2  and  an  algebraic  function  of  z.  We  thus  have  to 
consider  only  the  integrals  70?  I\,  .  .  .  ,  In-2  and  HI  =  /_1?  since  I-i  is 
a  special  case  of  HI,  viz.,  when  6  =  0. 

If  6  is  a  root  of  the  equation  R  (z}  =  0,  then  the  term  with  77jt+1  drops 
out.  Since  R(z)  cannot  have  a  double  root,  as  otherwise  it  could  be 
taken  from  under  the  root  sign  in  V  7?  (2),  we  may  in  this  case  express 

HI  through  the  integrals  770,  77_!,  .  .  .  ,  7/_(n_2^,    -  ~;  and  conse- 

2  —  6 

quently  through  integrals  of  the  type  7*.-  alone. 

ART.  164.     We  have  therefore  to  consider  the  integrals 


r  z*dz 

I     ,  - 

J  V. 


T 
Ik 

where  k  =  0,  1  .....  n  —  2,  where  n  is  the  degree  of  the  integral  func 
tion  R(z),  and  in  addition  the  integral 


184  THEORY   OF   ELLIPTIC    FUNCTIONS. 

where  6  is  a  root  of  the  equation  g(z)  =  0.  We  note  that  there  are  as 
many  integrals  of  the  type  HI  as  there  are  distinct  roots  of  the  equation 
g(z)  =  0.  The  quantity  6  is  called  the  parameter  (Legendre,  Functions 
Elliptiques,  t.  I,  p.  18)  of  the  integral  HI. 

ART.  165.  For  the  elliptic  integrals,  if  n  =  4,  we  have  the  integrals 
/o,  /i,  /2,  HI',  if  n  =  3,  there  are  the  integrals  70,  /i,  HI.  In  the  first 
of  these  cases  we  shall  see  that  /i  reduces  to  elementary  integrals;  and 
with  Legendre  we  call 

70  =    /  — — — -  an  elliptic  integral  of  the  first  kind, 
J  VR(z) 

/O     7 
z    Z    an  elliptic  integral  of  the  second  kind, 
\/R(z) 

H\=    / — an  elliptic  integral  of  the  third  kind. 

J  (z-b)  VR(Z) 

LEGENDRE'S  NORMAL  FORMS. 
ART.  166.     In  the  expression 

dz  dz 


VR(z)      VA(z  -  en)  (z  -  a2)  (z  -  a3)  (z  -  a4) 

let  us  make  the  homographic  transformation 

...  _  at  +  b 

ct  +  d  ' 
It  follows  that 

z  —  ak  =  — 


ct  +  d 
and  ad  - 


(ct  +  d)2 
We  then  have 

dz 
^  (ad  -  bc}dt 


We  note  that  the  expression  under  the  root  sign  is  not  essentially 
changed,  since  we  still  have  an  integral  function  of  the  fourth  degree, 
the  branch-points,  however,  being  different. 

Legendre  *  conceived  the  idea  of  so  determining  the  constants  a,  b,  c,  d 
that  only  the  even  powers  of  t  remain  under  the  root  sign.  If  we  neglect 
the  constant  A,  the  radicand  may  be  written 

[got2  +  git  +  g2][h0t2  +  hit  +  h2], 
*  Legendre,  loc.  cit.,  Chap.  II. 


ELLIPTIC   INTEGRALS   IN   GENERAL.  185 


where  go  =  (a  —  cai)  (a  —  ca2), 

g\  =  (d  —  cai)  (b  —  da2)  -f-  (a  —  ca2)  (b  —  ddi), 
,  g2  =  (b  —  dd\)  (b  —  dd2)j 

and  where  h0,  hi,  h2  are  had  when  we  interchange  a!  with  a3  and  a2  with 
a4  in  the  expression  for  the  g's. 

That  the  coefficients  of  t3  and  t  disappear,  we  must  have 

hog  i  +  go^i  =  0, 
gih-2  +  hig2  =  0. 
These  two  equations  are  satisfied  if  we  put 

0i  =  0  and  hi  =  0. 
From  the  expression  gi  =  0  it  follows  that 

2  ab  —  (dd  +  be)  (di  +  d2)  +  2cddid2  =  0; 
and  from  hi  =  0  we  have 

2  db  -  (dd  +  be)  (a3  +  d4)  +  2  cda3a4  =  0. 
These  two  equations  may  be  written 

„  \    i   o  d  c  n 

'i  T  &2)  ~r  £  -  -  did2  =  u. 
6  a 

i3  +  a4)  +  2  -  —  a3  &4  =  0. 
6  a 

From  them  we  may  determine  -  +  -  and  - .  -  considered  as  unknown 
quantities. 

If  d3  +  a4  =  a  i  +  d2  and  a3  •  a4  =  ax  •  a2,  the  two  equations  reduce 

to  one  and  then  we  need  only  determine  the  quantities  -  +  -  and  -  •  - 

b      a          b     d 

so  that  they  satisfy  the  one  equation.     When  these  two  quantities  have 

been  determined,  the  quantities  -  and  -  may  be  found  from  a  quadratic 

o          a 
equation. 

When  these  conditions  have  all  been  satisfied,  then  in  the  expression 


the  coefficients  of  t  in  both  factors  drop  out. 
We  have  finally 

dz  (ad  -  bc)dt 

g2)  (h0t2  +  h2) 


Legendre  further  wrote     ^  =  -  p2,         =  -  q2 

g-2  h2 

80  that  dz  (ad  -  bc)dt 


VR(z)       VAg2h2(l  -  p2t2)  (1  -  q2t2) 


186  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  finally  we  write  t  =  —  (the  Gothic  z  being  a  different  variable  from 

P 
the  italic  z),  we  have 

-  (ad  -  bc)dz 
dz  p ___ 


If  we  put  £=k2    and    C  =    ad 


the  above  expression  is 


-  z2)  (1  - 

The  quantity  k  is  called  the  modulus  (Legendre,  loc.  cit.,  p.  14).  In  theo 
retical  investigations  it  may  take  any  value  whatever,  real  or  imaginary; 
but  in  the  applications  to  geometry,  physics,  and  mechanics  we  shall  see 
in  the  Second  Volume  that  it  is  necessary  to  make  this  modulus  real  and 
less  than  unity. 

ART.  167.     If  we  make  the  above  substitutions  the  general  integral 
of  Art.  160 

CQ(z)dz     u 

•  -v\  /         becomes 


J  VR(z)  J  V(l  -  z2)  (1  -  k2z2) 

where  /(z)  denotes  a  rational  function  of  z.      We  may  write  this  function 
in  the  form 

,,    ,_    0(Z2)   +Z(/>T(Z2) 


where  <£,  <j>i,  ^,  ^i  denote  integral  functions.  If  we  multiply  the  numer 
ator  and  the  denominator  of  this  last  expression  by  ^(z2)  —  z^1(z2)f  it  is 
seen  that  /(z)  =  /0(z2)+  z/^z2),  where  /0  and  fi  are  rational  functions 
of  z. 

The  above  integral  correspondingly  becomes 

r  _  f(z)dz         =  r      Mz^dz    _  +  r      zf^z^dz 

J  V(l-  z2)  (1  -  /b2z2)      J  V(l  -  z2)  (1  -  A;2z2)      J  V(l  -  z2)  (1  -  Fz2) 

The  second  integral  on  the  right-hand  side  may  be  reduced  to  elementary 
integrals  by  the  substitution  z2  =  £. 

Proceeding  as  in  the  general  case  above  and  noting  that 


V(l-z2)(l-A;2z2) 
and 


d  ia-z^g-Fz2)"!  =  q0  +  ai(z2  -  6)  +  a2(z2  -  b}2  +  a3(z2 
(z2  -  b)k          J  (Z2  _  6)fc+i  \/(l-z2)(l-/c2z2) 


ELLIPTIC   INTEGRALS   IN    GENERAL.  187 

it  may  be  shown  that  the  integral 


is  dependent  upon  the  evaluation  of  the  integrals 

r dz t    r        z2dz 

J  Vl-z2l-£2z2'    J  V(l-z2)(l-A; 


These  integrals  are  known  as  Legendre's  normal  integrals  of  the  first,  second, 
and  third  kinds  respectively. 

ART.  168.     The  name  "  elliptic  integral  "  is  clue  to  the  fact  that  such  an 
integral  appears  in  the  rectification  of  an  ellipse.     Writing  the  equation 

cy  fy 

of  the  ellipse:  ^  -f  ^~  =  I,  the  length  of  arc  is  determined  through 


rx*  /-,  -  AM2j     r\  a4  - 

s  =    I     V  l  i~    j    }  dx  =   I     V  ; 

Jo    '  \<H7  J0   v        a' 


If  the  numerical  eccentricity  is  introduced: 


- 


/«      /a2  _  ^2  ^x  g2  _   g2j.2 

=    I     V/~; r-«—    I        /,   ,         9,  ,   2         2  ^dx. 

Jo   *     a2  —  ^2  Jo  v(a2  -  x2)  (a2  —  e2z2) 

If  further  we  put  x  =  a  sin  0,  it  is  seen  that 

s=   I    Vl  -  e2  sin2  (j>  d<f). 

i/O 

This  is  also  taken  as  a  type  of  normal  elliptic  integral  of  the  second  kind,* 
being  in  fact  composed  of  the  normal  forms  of  the  first  and  second  kinds 
as  above  defined.  Regarding  the  forms  of  the  integral  of  the  second 
kind  see  Chapter  XIII. 

ART.  169.     If  the  integral  which  we  have  to  consider  is  of  the  form 

f(z)dz 


where  f(z)  again  denotes  a  rational  function  of  z,  we  may  by  writing 

z  =  mt  -f  n 

make  a23  +  3bz2  +  3cz  +  d  =  ±t*-g2t-  g3, 

where  g^  and  g%  are  constants. 

This  is  effected  bv  writing   n  = ,  am3  =  4. 

a 

*  The  elliptic  integral  of  the  second  kind  was  considered  by  the  Italian  mathema 
tician  Fagnano  (1700-1766)  and  was  later  recognized  as  a  peculiar  transcendent  by 
Euler  (in  1761). 


188  THEORY   OF   ELLIPTIC    FUNCTIONS. 

The  above  integral  then  becomes 

F(t)dt 


r      F(I 

J  \/4  P  - 


g2t  -  93 

where  F(t)  is  a  rational  function  of  t.      The  evaluation  of  this  integral 
(cf.  Art.  165)  depends  upon  that  of  the  three  typical  integrals 


r        dt  r       tdt  r 

J  V4<3  -     t  -    3     J  V4P  -  g2t  -  93  '    J 


-  g2t  -  93     J  V4P  -  g2t  -  93          (t  -  b)  V4  13  -  g2t  - 

which  correspond  to  the  normal  forms  employed  by  Weierstrass. 
ART.  170.     In  the  expression 

(1)  R(z)  =  A(z  -  ai)  (z  -  02)  (z  -  a3)  (z  -  a4), 

make  the  homographic  transformation 

•-££•'./•.  ."/: 

and  so  determine  the  coefficients  *  that  to 

z  =  0,1,      z  =  a2,     z  =  as,     z  =  a4 
correspond 

i--!,    «  —  1,    i-4-l,    z  =  +  |. 

It  follows  immediately  from  (2)  that 


(5)  .-^-,  (0) 


where  p,  9,  r,  s  are  constants  which  may  be  determined  as  follows:  In  (4) 
write  z  =  a3,  z  =  1,  and  in  (5)  put  z  -  a2,  z  =  --  1.     We  thus  have 

2  2r 


-  a2  =   —  —  > 

1    -   fJL  1    + 


Equations  (4)  and  (5)  thereby  become 


z 


a3        1  +  /£      1  -  z 

.  i  —  — * —    •    • 


0.3-0,2          2         1  -  //z  a2  -  a3          2         1  -  ,«z 

In  the  same  manner  we  derive  from  equations  (3)  and  (6)  the  following: 

l--fl  l+£ 

#4  —  a\          2        1  —  «z  ai  —  a4          2        1  —  /tz 

*  Koenigsberger,  Elliptische  Functionen,  p.  271. 


ELLIPTIC   INTEGRALS   IN    GENERAL.  189 

Equations  (7)  and  (8)  become  through  division 

Z  —  Q2  =   fJL  —   1    .   1    +  Z 
2  —  a3         ,«  +   1        1   —   Z 

Writing  in  this  equation  z  =  a4,  z  =  -  ,  we  have 


q— 


-  +  1     A;  —  1 


and  similarly  for  the  values  z  =  ai,  z  =  —  -,  the  same  equation  gives 

K 
Ol   -  02  =    «  ~    1    ^  fc  ~    1 

ai  -  a3       «  +  1  '  k  4-  T 

The  quantities  A:  and  ,«  may  be  determined  from  the  last  two  expressions 
in  the  form 

(11) 


(12)  1  +  f*  =  «i  ~  «3  .  1  -  k 

1  —  a       a  i  —  0,2     1  4-  k 

From  the  equations  (9)  and  (7)  we  have 


(/z  2(1  -  («z)2  rfz  2(1  -  «z)2 

and  consequently 


O  /  1  -«\  '> 

2(1— /iz)a 

Through  the  multiplication  of  (7),  (8),  (9),  (10)  and   (13),  it  follows  at 
once  that 

dz 


r^^  =  JL 

J  x/7?^       M 


-Z2)(1-/C2Z2) 

where  M  =  - ( 


and  where  2  and  z  as  determined  from  (7)  and  (8)  are  connected  by  the 
relation 

_  a3  +  #2  _^  <*3  ~  a2     z  —  /£ 
2  2        *  1  -  az' 

the  quantities  u.  and  A:  being  determined  from  equations  (12)  and  (13). 


190  THEORY   OF  ELLIPTIC    FUNCTIONS. 

ART.  171.     If  in  the  equation 

we  put  the  right-hand  side  =  r,  then  the  six  different  anharmonic  ratios 
which  may  be  had  by  the  interchange  of  the  a's  are  denoted  by 

1.    ,  1       .          T  T-l. 

rV  '  1-r'  T-l'       r 

and  corresponding  to  each  of  these  values  there  are  two  values  of  k,  in  all 
twelve  values  of  k. 

Denoting  any  one  of  these  values  by  k}  it  is  seen  that  all  twelve  may  be 
expressed  in  the  form 


±k, 


i-k/         Vl-v  \i-ik 


(Cf.  Abel,  (Euvres,  T.  I,  pp.  408,  458,  568,  603;  Cayley's  Elliptic  Func 
tions,  p.  372.) 

Remark.  —  We  may  make  use  of  the  above  results  to  transform  the 
expression 

dz 


A(z  -  ai)(z  -  a2)(z  -  a3) 
into  Legendre's  normal  form. 

Noting  that  A(z  —  a\)(z  —  a2)(z  —  a3) 

-Limit    -—(z  -  ai)(z  -  a2)(z  -  a3)(z  -  a4)  1, 

a4  =  oo   L      a4  J 

we  have  to  write  in  the  formulas  above  —  -in  the  place  of  A,  and  let 

a4 

a4  become  infinite. 
We  then  have 


r  dz  J_  f  _= 

J  VA(z  —  a\)(z  -  a2)(z  —  a3)      M  J  \/ (\ 


dz 


-  k2z2} 
where  M  =  -  y  7        > 

_  a3  +  a2   ,   a3  —  a2      z  —  k 

Z    —  "  '  J 

and 


-  03 


ELLIPTIC   INTEGRALS   IN    GENERAL.  191 

ART.  172.     In  the  expression 


VR(z)  =  VA(z  -  ai)  (z  -  a2)  (z  -  a3)  (z  -  a4) 


write 


If  we  put  V(ai  -  o2)  (a i  -  a3)  .4=2  Af, 
we  have 


Choose  e  so  that  3  e  -  a'2  ~  °4  -  °3  ~  a4  -  0. 

ai  —  a2      ai  —  a2 


Let  ei  be  the  value  of  e  that  satisfies  this  equation,  and  write 

e 
We  finally  have 


&2   —  &4  i  &<*   — 

e2  =  e\  --  •  -  *•   and   63  =  e\  --  - 

0,1—0,2  a  i  — 


dz 


2  J/  ^(t-ei)(t-e2)(t-e3) 
.  dt 


where  eie2  +  62e3  + 

It  also  follows  that 


rP(z)dz  _   r 
J  \S          «/ 


where  P  and  p  denote  rational  functions. 

The  quantities  g2  and  g3  which  occur  in  Weierstrass's  normal  form  are 
called  invariants,  their  invariantive  character  being  especially  evidenced 
in  the  Theory  of  Transformation.  We  may  now  consider  more  carefully 
their  meaning. 

ART.  173.     Write  u  =   C    C^—> 

J  VR(z) 

where  the  function  R(z)  may  be  written 

R  (z)  =  a0z*  -f  4aiz3  +  6a222  +  4a32  +  a4. 


192  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Write  2  =  £!, 

x2 

dz  -  x*dxi  ~  ^1^2, 

X22 

where  the  variables  xi,  x2  individually  are  not  determined,  but  only  their 
quotient. 

We  then  have 

-f 


It  is  seen  that  /(a;  i;  x2)  is  a  binary  form*  of  the  fourth  degree.     We  have 
at  once 


If  next  we  write 

xi  =  <*>yi  +  by2} 

X2  =  cyi  + 


it  is  seen  that/(xi,  x2)  becomes  another  binary  form  </)(ylt  y2)  of  the  fourth 
degree. 

ART.  174.     In  general  make  the  above  substitutions  in  the  binary  form 
of  the  nth  degree 


,  x2)  =  a0^in  +  niaixin-*x2  +  n2a2x1n-2x22  +  •   -   •  +  anx2n, 
where  HI,  n2,  .  .  .  are  the  binomial  coefficients. 

We  thus  derive  another  binary  form  of  the  nth  degree  <j>(ylt  y2).     It  is 
seen  at  once  that 

—  /Oi,  x2)  =  a0zn  +  mai^-1  +  n2a2zn~2  +  ...+«„ 


=  a0(z  -  ai)  (z  -  a2)   ...   (z  -  an),    say. 
It  follows  that 


.  .   (xi  —  anx2), 
and  correspondingly 

<£(2/i,  2/2)=  ao'G/i  -  /?i?/2)  O/i  -  ft?/2)  -   •  .   (?/i  ~/?n?/2). 
Further,  since     xl  -  aix2  =  ayi  +  by2  -  «i(c?/i  +  d?/2) 


a  - 
*  Bocher,  Introduction  to  Higher  Algebra,  p.  260. 


ELLIPTIC    INTEGRALS   IN    GENERAL.  193 

it  is  evident  that  one  of  the  /?'s,  say 

and  similarly 

jj2=^d^bt    etc 
a  —  a2c 

From  this  it  is  clear  that,  if  some  of  the  a's  are  equal,  some  of  the  /?'s 
are  also  equal,  and  that  there  are  just  as  many  equal  roots  in  the  equation 
0(2/i,  2/2)  =  0  as  there  are  in/(xi,  x2)  =  0. 

ART.  175.     The  above  correspondence  gives  rise  to  the  following  con 
sideration:     Suppose  we  have  given  the  quadratic  form 

a0z2  +  2aiz  +  a2. 
The  roots  of  the  quadratic  equation 

aQz2  +  2  diz  +  a2  =  0 


are  z  = —  ±  — \  a\2  —  a0a2. 


If  we  write  ai2  —  a0a2  =  £>(ao>  ai>  a2),  we  know  that  the  two  roots  of 
the  quadratic  equation  are  equal  if  D  is  equal  to  zero.  The  quantity 
D  after  Gauss  is  called  the  discriminant  of  the  quadratic  equation. 

Also  for  forms  of  higher  order  we  may  derive  such  discriminants,  whose 
vanishing  is  the  condition  that  the  associated  equation  have  equal  roots.* 

The  quantity  D(a0,  ait  a2,  .  .  .  ,  an)  is  an  integral  rational  function 
of  a0,  0,1,  .  .  .  ,  an  and  is  homogeneous  with  respect  to  these  quantities. 

If  next  we  form  the  discriminant  D(a0',  a/,  a2,  .  .  .  ,  an')  of  the 
form 

,  2/2)  =  fl(/2/in  +  niaifyin~li/2  +  n2a2'ijin-2i/22  +  •   •   •  +  an'?/2n, 


then  the  vanishing  of  this  discriminant  is  the  condition  that  0(?/i,  2/2)  have 
equal  roots  /?.  But  we  saw  that  0(?/i,  y2)  had  equal  roots  when  the 
roots  off(xi,  x2)  are  equal.  It  follows  that 

D(a0',ai',  .  .  .  ,ow/)=CD(a0,ai,  .  .  •  ,  on), 
where  C  is  a  constant  factor.     This  constant  factor  *  is  {  ad  —  be  }  n(n~  x). 

*  Cf.  Salmon,  Modern  Higher  Algebra,  p.  98;  Burnside  and  Panton,  Theory  of  Equa 
tions  (3d  ed.),  p.  357;  etc. 

*  Cf.  Salmon,  loc.  cit.,  p.  108;  Bocher,  loc.  cit.,  p.  238. 


194  THEORY   OF   ELLIPTIC   FUNCTIONS. 


ART.  176.  If  the  function  f(xi,x2)  =  a0Xin+  n\aiXin-  Ix2  +  •  •  •  +  anx2n 
becomes  through  the  substitutions 

xi  \\ayi  +  by2, 
x2  \\cyi  +  dij2, 

</>(yi,  2/2)  =  a>o'yin  +  nialryin-ly2  +  •   •   •  +  an'y2n, 
and  if  I  is  a  function  of  the  coefficients  such  that 

/(«</,  a>i,  -  •  •   ,  On')  =  (ad  -  bc)^I(a0)al}  .  .  .  ,  an), 

where  /JL  is  an  integer,  then  /  is  called  an  invariant  of  the  form/(xi,  x2). 

It  may  be  shown  *  that,  if  /  is  an  invariant,  /*  must  be  equal  to  £  up, 
where  p  is  the  degree  of  /  with  respect  to  the  coefficients  a0,  «i,  .  .  .  ,  an. 
The  quantity  /JL  is  sometimes  called  the  index  of  the  invariant. 

The  following  theorem  is  also  truerf  All  the  invariants  of  a  binary 
formf(xi,  x2)  may  be  expressed  rationally  through  a  certain  number  of  them 
which  are  called  the  fundamental  invariants. 

For  the  form  of  the  fourth  degree, 


there  are  only  two  fundamental  invariants   (cf.  Sylvester,  Phil.  Mag., 
April,  1853). 

The  one  of  these  is  J 

72=  a0a4  —  4«ia3  4-  3a2. 

If  by  the  given  transformations  we  bring  f(x\,  x2)  to  the  form 

0(2/i,  2/2)  =  «o'?/i4  +  4a1/?/13!/2  +  -   -   -  +  a/2/24, 
then  it  is  easy  to  show  that 

ao'a4'  —  4  ai'oa'  +  3  a22  =  (a0«4  —  4  a^a^  +  3  a22)(ad  —  be)4. 


In  this  case  p  =  2,  w  =  4,  /JL  =  J  n/o  =  4. 
We  thus  have 

72'  =  72(ad 

The  other  fundamental  invariant  §  is 

73  =  a0a2a4  +  2a!a2a3  -  a23  - 

It  is  seen  at  once  that 

73'  -  73(ad  -  6c)«. 

*  Cf.  Salmon,  loc.  cit.,  p.  130;  Burnside  and  Panton,  loc.  cit.,  p.  376. 

f  Cf.  Salmon,  pp.  Ill,  132,  175;  Bocher,  loc.  cit.,  Chap.  XVII,  and  Burnside  and 
Panton,  p.  405. 

%  Salmon,  loc.  cit,  p.  112.  Cayley,  Cambridge  Math.  Journ.  (1845),  Vol.  IV,  p.  193, 
introduced  this  invariant. 

§  To  Boole,  Cambridge  Math.  Journ.  (1841),  Vol.  Ill,  pp.  1-106,  is  due  the  discovery 
of  this  invariant;  see  also  Cambridge  Math.  Journ.,  Vol.  IV,  p.  209;  Cambridge  and 
Dublin  Math.  Journ.,  Vol.  I,  p.  104;  Crelle,  Bd.  30,  etc.;  and  Eisenstein,  Crelle,  Bd.  27, 
p.  81;  Aronhold,  Crelle,  Bd.  39,  p.  140. 


ELLIPTIC    INTEGRALS   IN    GENERAL. 


195 


ART.  177.  The  discriminant  D  of  the  binary  form  f(x\,  x2)  may  be 
rationally  expressed  (cf.  Salmon,  loc.  cit.,  p.  112)  in  terms  of  72  and  73  in 
the  form 

D  =  723  -  27  732. 
It  is  evident  that 

D'  =  72'3  -  27  73'2  =  D(ad-  be)12. 

ART.  178.     The  functional-determinant  or  Jacobian  of  the  two  forms 
L>  X2),  ^2(^1,  xz)  may  be  written 


F  = 


dx2 

6^2 
dx2 


If  we  make  the  substitution 

*i  =  ^1(2/1,2/2),  %2  =  ^2(2/1,2/2), 

>l!  and  /12  being  functional  signs,  then  ^i(xi,  x2)  becomes  a  function  of 
?/i,  2/2,  which  may  be  symbolically  denoted  by  [^i(zi,  x2)]s  and  ^2(xi,  x2) 
becomes  by  the  same  substitution  [^2(xi,  x2)]s. 

We  form  the  functional  determinant  of  these  two  forms 


and  we  shall  study  the  relation  between  F  and  <1>. 
It  is  evident,  since 

ch/r   =  6^r    6X!         3 

dz/i       dxi  d#i       a 
that 


L 


'iOi,:r2)1  a^i  .  ^^1(^1^2)1  &k 

I rl  — ' '   I 

dx\        js^l/2  dx2        J$a?/< 


k 


— L    L 


Suppose  next  that 
We  then  have 


=  AI(I/I,  2/2) 
=  k(y\,  2/2) 


-  in 


a,  6 


a?/2 


+  by 


-  6c). 


196 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


ART.  179.     Let/(#i,  x2)  be  a  binary  form  of  the  nth  degree.     It  is  seen 

that  df(xij  x<^  and  dKXl>  X2}  are  binary  forms  of  the  degree  n  -  1. 

dxi  dx2 

The  functional-determinant  F  of  these  two  functions 

axiax2 

jy 

ax22 


=  H(f),  say, 


is  called  the  Hessian  covariant  *  of  the  form  /.     Suppose  that  by  the 


substitution 


Xl  =  ayi 


+ 


the  function /(zi,  x2)  becomes  <f>(yit  y2)  and  form  the  Hessian  covariant 
for  this  latter  function,  viz., 

fyi  \dyij         dy2  \diji/ 


d_/d±\ 

)?/2  \dyal 


We  have 

or 

and  similarly 

When  these  values  are  substituted  in  the  above  determinant,  it  follows  that 
a  - 


fyi 


=  (ad  -  be) 


9^2 


Further,  since  — 


a  +    i —    c,  etc.,  we  have 


+  d 


7,1 


*  Cf.  Salmon,  Zoc.  ci«.,  p.  117. 


ELLIPTIC   INTEGRALS   IX   GENERAL.  197 

It  follows  that 


=  (ad  —  be)2 

and  consequently 

#(<£)  =  (ad-l 

ART.  180.     We  may  consider  more  closely  the  meaning  of  the  covariant. 
Suppose  we  have  a  binary  form  f(x\,  x2)  of  the  nth  degree.     With  its 
coefficients  a0>  ai,  -  •  •  >  an  and  with  x\,  x2  we  form  an  expression 
/"» (  ) 

Cj#o>al>    •   •    •    i    ton')    %1,%2\> 

C  denoting  a  functional  sign  which  with  respect  to  x\,  x2  is  of  the  vth 
degree,  and  in  regard  to  the  a's  it  is  of  the  ^th  order. 
Suppose  further  that  by  the  substitution 

xi  \\ayi  +  by2, 
%2  \\cyi  +  dy2, 

the  function/(xi,  x2)  becomes  </>(yi,  y2}. 

With  the  coefficients  a0',  a\  ,  .  .  .  ,  an'  of  <f>(yi,  y2)  and  with  t/i,  y2 
we  form  the  same  function 

/"»(//  / .  ) 

L  {a0  ,  a\  ,  .  .  .  ,  an  ,  y\,  y2j. 


If  then 


,an'; 


wrhere  ,«  =  ?(np  —  v), 

we  say  that  C  is  a  covariant  *  of  the  binary  form/(xi,  x2). 

ART.  181.  In  the  theory  of  covariants  it  is  shown  that  for  every  binary 
form  f(xi,  x2)  there  is  a  finite  number  of  independent  covariants,  through 
which  all  the  other  covariants  may  be  expressed.^ 

If  f(xi,  x2)  is  a  binary  form  of  the  fourth  degree,  say 

f(xi,  x2)  =  a0xi4  -f  4ai^i3.r2  +  6a2Xi2x22  +  4a3Jix23  4-  a4x24, 

there  are  two  fundamental  covariants   (Salmon,  loc.  cit.,  p.   192):     The 
one  is  the  Hessian,  where 

v  =  n-2  +  n-2  =  2n-4,     p  =  2; 
and  consequently 


*  Salmon,  loc.  cit.,  p.  135;  Burnside  and  Panton,  loc.  cit.,  p.  376. 
t  Salmon,  loc.  cit.,  pp.  132,  175,  176;  and  see  also  Clebsch,  Theorie  der  binaren  alge- 
braischen  Formen,  pp.  255  et  seq. 


198  THEORY   OF   ELLIPTIC   FUNCTIONS. 

This  covariant  is 


2(0,10,4  —  a2a3) 


2ala3-3a22) 
—  a32)  x24. 


The  other  fundamental  covariant  is  the  Jacobian  of  the  quartic  and  its 
Hessian: 


r-I 


dx 


dx2 

dH(f 

dx2 


and  therefore 
so  that 


For  this  covariant  it  is  seen  that 

t     "  *  t 

T'  =  [T]s  (ad  -  be)*. 

ART.  182.  Between  the  two  co variants  T  and  H(f)  there  exists  the 
relation  * 

-  T2  =  73/3  -  I2f2H(f)  +  4#(/)3- 

This  formula  is  given  by  Cay  ley  in  Crelle's  Journal  (April  9,  1856, 
Bd.  50,  p.  287).  The  formula,  however,  as  stated  by  Cay  ley,  is  due  to  a 
communication  from  Hermite.f 

We  have  at  once 


or  writing -    =  £,  it  is  seen  at  once  that 


ART.  183.     Consider  next  the  determinant 


H(f),f 
dH(f),  df 


dx 


H(f), 
dx\  - 


f 


dx, 


dxl 


dx, 


The  functions/  and  H(f)  being  homogeneous  of  the  fourth  degree  in 
,  x2)  it  follows  that 


dxi  dx2 

*  Cf.  Salmon,  loc.  cit.,  p.  195;  Halphen,  Fonctions  Elliptiques,  t.  II,  p.  362;  Clebsch, 
loc.  cit.,  §62. 

t  Similar  relations  have  been  derived  by  Hermite  for  the  quintic  and  for  every 
form  of  odd  degree  (cf.  Salmon,  p.  249). 


ELLIPTIC   INTEGRALS   IN   GENERAL. 
We  therefore  have 


dJTi 
*L 

dxl 


199 


dx2 

«L 

dx2 


dx2 


ax 


*2 


dx 


=  2T(x2dxi-Xidx2). 


On  the  other  hand 

A  - 

It  follows  that 
or 


H(f),        f 
dH(f),     df 


=  H(f}df-fdH(f} 


=  -2  T(x2dxl  - 

—  —  4  (x2dxi  — 

4 

~  (-  2)* 
4 


- 


(-2)* 


(2/3  -  72C  -f 


From  this  it  is  evident  that 
x2dxl 


(-2)* 


Since  z  =  — ,  it  follows  that 


where 

We  finall    have 


,  x2)  \  5( 

=  a0z4  +  4  a^3  4-  6  a2z2  +  4a3z  +  a4. 


r   ^^    =  (-2)*  r 

J  \  R(z]          -  4   J  \ 


3  - 


This  is  practically  the  transformation  given  by  Cayley  *  in  Crdle  ~ 
Journal,  Bd.  55,  p.  23. 

*  See  also  Cayley,  Elliptic  Functions,  p.  317;  and  Burnside  and  Pant  on,  loc.  cit., 
p.  474;  Brioschi,  Sur  une  formule  de  M.  Cayley,  Crelle,  Bd.  53,  p.  377,  and  Crelle,  Bd.  63, 
p.  32.  The  Berlin  lectures  of  the  late  Prof.  Fuchs  have  been  of  great  assistance  in 
the  derivation  of  this  transformation. 


200  THEORY   OF   ELLIPTIC    FUNCTIONS. 

The  mode  of  procedure,  however,  as  noted  above,  was  suggested  by 
Hermite  (cf.  Her  mite  in  "Lettre  123  "  of  the  Correspondence  d'Hermite  et  de 
Stieltjes;  read  also  letters  124  and  125  of  the  above  correspondence  and 
Hermite,  Crelle,  Bd.  52;  Cambridge  and  Dublin  Math.  Journ.,  vol.  IX, 
p.  172;  and  t.  I  of  the  Comptes  Rendus  for  1866). 

If  we  write  2  t  for  £  in  the  above  formula,  it  becomes 


r  dz    =  _  .  r         dt 

J  VR(z)  J  2V4t*  -  I2t  +  / 


ART.  184.     Weierstrass  employed  a  somewhat  different  notation.     He 
put 

^2  =  92,         /3  =  —  93, 

and  consequently  introduced  as  his  normal  form  of  the  elliptic  integral  of 
the  first  kind. 


f: 


He  further  wrote 

4*3  -g2t-g3  =  ±(t-  ei)  (t  -  e2)  (t  -  e3)=S(t), 

so  that  (cf.  Art.  172)  between  the  e's  and  the  0's  we  have  the  following 
relations: 

ei  +  e2  +  e3  =  0, 


ART.  185.     We  may  show  as  follows  how  the  Hermite-  Weierstrass  nor 
mal  form  may  be  brought  to  the  Legendre-Jacobi  normal  form. 
In  the  expression 

dt 


T) 

write  t  =  A  +  —  ,  where  A  and  B  are  constants.     It  is  seen  at  once  that 
z2 

dt  -  Bdz 


VS(t) 


Under  the  root  sign  there  is  an  expression  of  the  sixth  degree  which  con 
tains  only  even  powers  of  t.     But  by  writing 

A  =  63, 
this  reduces  to 

dt  -  Bdz 


VS(t)       VB{(e3  -  ei)z*  +  B}  \  (e3  -  e2)22  +  B} 


ELLIPTIC    INTEGRALS   IX    GENERAL.  201 

If  further  we  give  to  B  the  value 

B  =  ei  -  e3, 

and  put 

e2  —  e3  _  7  2 

-    K    , 

e1  -  e3 
wehave  dt  1  dz 


VS(t)          Vei  -  e3  v(l-z2)(l-A:2z2) 
It  has  thus  been  shown  that  through  the  substitution 


the  Weierstrassidn  normal  form  is  changed  into  that  of  Legendre. 

Other  methods  of  effecting  this  transformation  will  be  found  in  Volume  II. 

ART.  186.     If  we  write  *  l 

€l  -  e3  =  -, 

£ 

then  is  1 

ei  =  -  +  e3.  (1) 

£ 

Further,  since  k2 

e2  -  e3  =  —, 

we  have  1.2 

e2  =  j  +  e3;  (2) 

and  using  the  relation 

ei  +  e2  +  e3  =  0, 
we  also  have  -,   -,    ,   T  2 

1    1   ~r  K 


This  value  of  e3  written  in  (1)  and  (2)  gives 


e2«_  (2*2-1).  (5) 

O  £ 

From  the  equations  eie2  +  e2e3  +  e3e\  =  —  \g2  and  e\e2e3  =  J  g3  it  follows 
with  the  use  of  (3),  (4)  and  (5)  that 

-     e202  =  (2  -  *2)  (2  *2  -  1)  -  (2  -  A-  2)  (1  +  A-2)-  (2  A*  2  -!)(!+  A:2), 


. 

and  from  these  two  relations  t 

^23  108J1  -A-2  +  A-4}3 


*  Cf.  Halphen,  Fonctions  Elliptiques,  t.  I,  p.  25. 

t  Cf.  Felix  Miiller,  Schlomilch  Zeit.,  Bd.  18,  pp.  282-287. 


202  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  shall  next  show  that  the  above  expression  is  an  absolute  invariant* 
that  is,  it  remains  entirely  unchanged  by  a  linear  substitution. 
We  have 


and  we  saw  in  Art.  176  that 

/2/3  -  /23M  - 

and  73/2  =  /32  (ad  -  be)12. 

It  follows  that  3      j  3      j  /3 

_   2          72          J  '  2,' 
93  +3  *3 

From  this  it  is  seen  that  k  is  the  root  of  an  algebraic  equation  of  the  12th 

T  3 

degree  whose  coefficients  depend  rationally  upon  the  absolute  invariant  ^2-- 

ART.  187.     Riemann's  Normal  Form.-\     If  in  Legendre's  normal  form 

dz 

V(l-z2)(l-/c2z2) 
we  put  z2  =  t,  k2  =  X,  it  becomes 

1    r  dt 


v  t  (1  -  0  (i  -  Jit) 
If  in  the  latter  integral  we  write 


we  have,  neglecting  a  constant  multiplier, 

dr 


Vt(l  -  pr  +  T2) 

(Kronecker,  BerZm  £tte.,  July,  1886.) 

In  Volume  II  the  transformation  of  the  general  integral  into  its  normal 
forms  will  be  resumed  and  the  discussion  for  the  most  part  will  be  restricted 
to  real  variables. 

ART.  188.  In  connection  with  the  realms  of  rationality  we  may  consider 
more  closely  the  integrals  that  have  been  introduced  in  this  Chapter. 

Let  R  denote  any  rational  function  of  its  arguments,  and  write  the 
integral 


where  a  —  \/az  +  6.     If  we  put  a  =  ^(2),  where  ^r  is  a  rational  function, 
then  z  —  (j)(t)  is  a  rational  function.     The  above  integral  becomes 


*  Salmon,  loc.  dt.,  p.  111. 

t  Cf.  Klein,  Math.  Ann.,  Bd.  14,  p.  116,  and  Theorie  der  Elliptischen  Modulfunc- 
tionen,  Bd.  I,  p.  25. 


ELLIPTIC   INTEGRALS   IX    GENERAL.  203 

where  the    integrand  is    a   rational  function    of  t.      For    example,   put 

,9  7 

a  =  Vaz  +b  =  t,  then  z  =  •     In  .this  case  the  realm  (z,  a]  is  evi 

ct 

dently  the  same  as  the  realm  (t),  since  (z,  t)  is  the  same  as  (t),  the  pres 
ence  of  z  within  the  realm  adding  nothing  to  it,  as  z  is  a  rational 
function  of  t. 

Consider  next  the  integral 

*,  o)dz, 


where  a  =  \/(z  —  a\)(z  —  a2)  =  (z  —  a2)  \  -  -  • 

*   z  —  0,2 

By  writing  t2  =  z  ~  Ol,  it  is  seen  that  a  =  (z  -  a2)t  and  z  =  ^  . 

z  —  a2  I  -  t2 

We  note  that  both  a  and  z  are  rational  functions  of  t  and  that  t  is  a 
rational  function  of  o  and  z.  Hence  every  rational  function  of  a  and  z  is 
a  rational  function  of  t  and  any  rational  function  of  t  may  be  expressed 
rationally  through  z  and  a.  In  this  case  we  may  say  that  the  two  realms 
(z,  a)  and  (t)  are  equivalent  and  write 

(z,  o}  ~  (0. 
In  the  case  of  the  integral 


J  R(x,  Vax2  +  2  bx  +  c)dx, 


if  we  put  y2  =  ax2  -f  2  bx  +  c,  we  have  the  equation  of  a  conic  section. 
This  conic  section  is  cut  by  the  line 

y  -  r,  =  t(x  -  c), 

where  t  is  the  tangent  of  the  angle  that  the  line  makes  with  the  x-axis, 
at  the  point  c,  y,  say,  and  at  another  point 

26  -  2rt  +  &2 


t*  -  a 

-  2a*t  -  2bt 


r  —  a 
Hence  as  above 

(*,  y}  -  (0. 

In  the  case  of  the  integral 


where  s  is  the  square  root  of  an  expression  of  the  third  or  fourth  degree  in 
z,  it  was  shown  by  both  Abel  and  Liouville  that  the  integrand  cannot  be 
expressed  as  a  rational  function  of  t.  This  we  know  a  priori  from  our 
previous  investigations;  for  we  saw  that  an  elliptic  integral  of  the  first 


204  THEORY   OF   ELLIPTIC    FUNCTIONS. 

kind  nowhere  becomes  infinite,  while  the  integral  of  a  rational  function 
must  become  infinite  for  either  finite  or  infinite  values  of  the  variable. 

In  Art.  166  it  is  seen  that  z  and  s  may  be  rationally  expressed  through 
z  and  s  =  \/(l  —  z2)(l  —  /c2z2)  and  at  the  same  time  z  and  s  may  be  ration 
ally  expressed  through  z  and  s  so  that 

(z,  s)  ~  (z,  s), 

and  consequently  any  element  of  one  realm  is  an  element  of  the  other. 
It  is  also  seen  that  if  r  =  \/4  t3  —  g2t  —  g%,  then 


We  note  that  by  these  transformations  the  order  of  the  Riemann  surface 
remains  unchanged. 

The  above  three  realms  of  rationality  being  equivalent,  the  name  elliptic 
realm  of  rationality  may  be  applied  indifferently  to  them  all. 

EXAMPLES 

1.   In  the  homographic  transformation, 

a  .  +  /ft  +  yz  +  dtz  =  0 
for  fc       z  =  flj,        z  =  a2,        z  =  a3,        z  =  a4, 

let  t  =  0,          t  =  1,          t  =  -,          t  =  oo. 

We  thus  have 

a  +  yal  =  0)        a  +  /?  +  -ya2  +  da2  =  0,        «A  +  /?+  7^3  +  ^«3  =  0, 
The  vanishing  of  the  determinant  of  these  equations  gives 


ai  —  03     a2  ~  a4 
Show  that  —  -  is  thereby  transformed  into  Riemann  's  normal  form. 


2.   In  a  similar  manner  transform  —  -  into  Legendre's  normal  form  and  from 


the  resulting  determinant  derive  the  12  values  of  k  given  in  Art.  171.     [Thomae.] 
3.   Show  that  the  substitutions 

z~ai    a*~a*       .2      QS  -  <*4    "2  ~  «i 


transform 


into 


±v^(o4-aa)(a1-as)   /    — = 
Jai  \/(z- 


-  Z)  (1  -  k2£)  JaiV(z-  aj)  (z-  a2)  (z-  ag)  (2-  o4) 

[Riemann-Stahl,  ^/.  ^wnd.,  p.  16.] 


ELLIPTIC   INTEGRALS   IX   GENERAL.  205 

4.   Show  that  the  substitution 


z  -  a2  '  03  - 
transforms 


/; 


into    '  * 


VA(z  -  Ol)(2  -  a2)(2  -  as)(2  -  a4)  t/  >/4(U  -  a^^  -aj(t-  aj(t  -  a4) 

[Burkhardt,  Ett.  Fund.] 
Derive  two  other  such  substitutions. 

5.   Show  that  the  substitution 

t  =  e    +  ^  -gi)(g3-gi) 


transforms  Weierstrass's  integral  into  itself. 

6.   If  a  is  a  root  of  az3  +  3  br  +  3  cz  +  d  =  0,  by  writing  z  -  a  =  z2  transform 

=1^=^=  into  Legendre's  normal  form. 
3 


7.   If  f(x)  =  x4  +  6  mx2  +  1,  show  that 

4  /•-=*  <*  * 

•/  vV  + 


where 


x4+  6mx2+  1  / 

[Appell  et  Lacour,  Fonc.  Ellip.,  p.  268.] 


CHAPTER  IX 

THE  MODULI  OF  PERIODICITY  FOR  THE  NORMAL  FORMS  OF 
LEGENDRE  AND  OF  WEIERSTRASS 

ARTICLE  189.     The  Riemann  surface  for  the  elliptic  integral  of  the  first 
kind  in  Legendre's  normal  form, 

'-^L,  where  Z  =(l  -  z2)(l  -  /c2z2)=  s2 

VZ 

has  the  branch-points  +  !,—!,  +  ->  —  -• 

A/  /V 


4-co 


In  the  figure  *  we  join  the  points  +  1  and  —  1  with  a  canal  and  also 
the  points  +  -  and  —  -  with  a  canal  which  passes  through  infinity.     Here 

K  Ki 

we  have  taken  the  modulus  k,  which  may  be  any  arbitrary  complex 
quantity,  as  a  real  quantity,  positive  and  less  than  unity.     In  the  follow 
ing  discussion  we  make  no  use,  however,  of  this  special  assumption. 
In  Art.  142  we  saw  that 


b  8 

The  corresponding  quantities  here  are,  say, 


Vz 

and  B(k)=2f+1-^. 

J  -  i  VZ 

*  Cf.  Koenigsberger,  Ellipt.  Fund.,  pp.  299  et  seq. 
206 


MODULI   OF   PERIODICITY.  .  207 

For  any. integral  in  the    T'-surface  we  shall  take  as  lower  limit  the 
point  z0  =  0,  SQ  =  4-  1 ;  that  is,  the  origin  in  the  upper  leaf. 
We  then  have 


«(z,.)-.|      ^%inr. 

*/o,i  vZ 

If  we  let  the  upper  limit  coincide  also  with  the  point  0,  1,  then,  however 
the  curve  be  drawn  in  the  T'-surface,  we  have  always 

(I)  u(0,  1)  -  0. 

ART.  190.     In  Art.  139  we  saw  that 

on  the  canal  a,  u(X)  —  u(p)  =  A(k), 
and  on  the  canal  b,  u(p)  —  u(X)  =  B(k). 

We  form  the  integral  between  arbitrary  limits,  Z2,  S2  and  z\,  s\,  where 
the  path  of  integration  is  free,  that  is,  taken  without  regard  to  the  canals 
a  and  b. 

If  the  path  of  integration  crosses  the  canal  a  (see  Fig.  63)  we  have 


/'Zj.Sj  f>p  ^»X  /»Zi,S! 

«/Z2,S2  t/Z2,S2  *J  p  */ A 

the  integrand  for  all  these  integrals  being 


-  z*)(l  -  fc*z*) 

Noting  that  the  second  integral  on  the  right-hand  side  is  indefinitely 
small  in  T,  it  is  seen  that 

_^  =  u(p)  -  i7(z2,  32)  +  u(zlf  Si)  -  M(X)  in  T 

VZ 


=  M(ZI,  Si)  —  ^^(z2,  s2)  —  A(k). 
If,  however,  the  integration  is  taken  in  the  opposite  direction,  we  have 

/*Z2,S2    ^^  _  _ 

<        — =  =  u(zo.  So)  —  u(zi,  Si)  +  A.(k). 

J***  \/z 

We  may  form  the  following  rule :  //  the  path  of  integration  for  the  integral 

dz 


-  z2)(l  -  £ 

crosses  the  canal  once  in  the  direction  from  p  to  X,  this  integral  with  free  path 
is  equal  to  the  integral  taken  in  T'  decreased  by  A(k);  but  if  we  cross  the  canal 
a  in  the  direction  from  X  to  p,  then  this  integral  with  free  path  is  equal  to  the 
integral  in  T'  increased  by  A(k).  Upon  crossing  the  canal  b  we  have  the 
oppos-ite  result :  If  b  is  crossed  in  the  direction  from  p  to  X,  then  B(k)  is  to  be 
added  to  the  integral  in  T' . 


208  THEORY   OF  ELLIPTIC   FUNCTIONS. 

We  may  apply  this  rule  in  order  to  derive  a  number  of  formulas,  which 
give  the  value  of  u(z,  s)  at  certain  points.  In  Fig.  63  it  is  seen  that  in 
the  upper  leaf  of  T' 


But  in  the  lower  leaf  where  the  path  of  integration  is  taken  congruent  to 
the  one  in  the  upper  leaf,  there  being  no  canal  between  the  points  —  1 


and  -  -, 
k 


C~l  dz 

Jf  ,  7i 


If  we  add  these  two  integrals  and  note  that  the  elements  of  integration 
are  equal  in  pairs  and  of  opposite  sign,  it  is  seen  that  the  two  integrals  on 
the  left-hand  side  cancel,  so  that 


or 


Consider  further  the  integral  from  —  1  to  +  1  in  the  upper  leaf  and 
on  the  upper  bank  of  the  canal  from  —  1  to  +1  (the  upper  bank  being 
the  one  nearest  the  top  of  the  page) 


-i  VZ 
The  same  integral  in  the  lower  leaf  and  on  the  upper  bank  of  the  canal  is 


It  follows,  as  above,  that 


Next  forming  the  integral  from  +  1  to  +  -  in  the  upper  leaf  and  upper 
bank,  we  have 


+1  vZ 

and  in  the  lower  leaf,  upper  bank, 


MODULI   OF    PERIODICITY.  209 

We  therefore  have 


We  then  form  in  the  upper  leaf,  upper  bank, 

»+oo,+oo   J7 


k 

and  on  the  lower  leaf,  upper  bank, 

»+00, -00    fJ7 


Adding  these  two  integrals  we  have 

(V)  #(&)  =  7Z(oc,  +  oc)+ ?Z(cc,  —  oc)  —  2 

ART.  191.     If  we  form  the  integral 

1 dz 

V(l  -z2)(l  -  A-2z2) 

in  the  upper  leaf  of  T'  and  take  the  integration  along  the  upper  bank 
of  the  canal,  it  is  seen  that  the  path  of  integration  is  congruent  to  the  one 
from  +  -  to  +  oo .  At  two  corresponding  points  of  the  paths  the  abso- 

*v          K 

lute  values  of  z  are  the  same,  but  the  signs  are  opposite.  This  difference 
of  sign,  however,  does  not  appear  in  the  expression  (1  —  z2)(l  —  k2z2). 
The  differential  dz  is  the  same  along  both  the  paths  and  positive,  and 
consequently  the  elements  of  integration  are  equal  in  pairs  and  we  have 


-  z2)(l  -  k2z2)     «/i    V(l  -  z2)(l  - 

In  a  similar  manner  we  have 

dz  C+*  dz 


J 


_  V(l  -  z2)(l  -  £2z2) 

We  form  the  integration  over  the  path  indicated  in  Fig.  64  which  lies* 
wholly  in  the  upper  leaf  and  passes  twice  through  infinity. 

The  integral  I  —  z  taken  over  this  path  must  be  zero. 

J  V'(l  -  z2)(l  -  £2z2) 

since  the  path  of  integration  does  not  include  a  branch-point. 


210 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


We  therefore  have 
.+  ? 


/+ 
1 


upper 
bank 


upper 
bank 


We  note  that  the  two  integrals 


J.i  Vz 

+k 

upper  bank 


upper     lower 
bank     bank 


lower 
bank 


VZ 

lower  bank 


=o. 


are  equal,  for  the  sign  of  dz  is  different  in  both  integrals,  and  as  both  inte 
grals  are  in  the  upper  leaf  but  upon  different  banks,  there  is  a  difference 


Fig.  64. 

in  sign  and  also  a  difference  in  sign  due  to  the  limits  of  integration.     On 
the  other  hand  the  two  integrals 

.  i 

dz 


f    ^ 

J+i   VZ 


and 

+i       Z 


i  VZ 


are  equal  with  opposite  sign,  since  there  is  no  canal  between  the  two 
paths  over  which  the  integration  is  taken. 
f  It  follows  that  the  sum  of  the  above  integrals  reduces  to 


i    VZ 

k 


-i  VZ 
where  the  integration  is  on  the  upper  bank  for  all  the  integrals. 


MODULI   OF   PERIODICITY.  211 

Owing  to  the  relation  (M)  above,  this  sum  of  integrals  further  reduces, 
after  division  by  2,  to 


-i  VZ        J+i    VZ 
It  follows  at  once  that 

or,  owing  to  (III), 


If  we  take  the  congruent  path  of  integration  in  the  lower  leaf,  we  again 
have,  since  no  canals  are  crossed, 


or          .  -      _ 

We  have  thus  the  formula 


ART.  192.  We  compute  the  integral  from  0  to  1  in  the  upper  leaf  of 
T'  on  the  upper  bank  of  the  canal  and  then  the  integral  taken  over  the 
congruent  path  in  the  lower  leaf. 

It  is  clear  that 


Jo 


ro,i   VZ      Jo,-iVZ 

upper  leaf  lower  leaf 

It  follows  that 


-)+ i7(+  1)-  w(0,  -  1)=0, 
or,  since  w(0,  1)  =  0  from  (I),  we  have 
(VIII)  2u(+  1)-  w(0,  -  l)  =  - 

Further,  it  is  seen  that 

r°-+1_rfz_  =   /"+1  _^z_ 
*/-i     VZ      Jo.+iVZ 

upper  bank     upper  bank 

and  consequently,  multiplying  by  2,  we  have 


-        =  2 
-i  v  Z  o,i  \  Z 

upper  bank      upper  bank 


212  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

From  this  it  follows  that 

u(+  1)-  Ti(-  1)+  B(k)=  2{u(+  1)  -  u(Q,  1)  - 
or,  owing  to  (I)  and  (III), 


We  thus  obtain 
(IX)  -$B(k)=u(+l). 

We  have  thus  derived  the  following  nine  formulas : 

(I)u(0,l)-0,  (V)5(oo, 

=  B(K), 

/TT\      —  /  -i\  ~~      /  J-\  -^J-  \  *v  )  /TTT\      —   /  \ 

(II)  u(-  1)  -  U  [--  )  =  —^r2'          (VI)  W(oo,+  oo)- 


(VII) 


From  these  formulas  we  have  at  once  : 


ART.  193.  Legendre  *  and  Jacobi  f  did  not  use  the  quantities  A  (k)  and 
B(k)  but  instead  two  other  quantities  K  and  Kf.  These  quantities  are 
connected  with  A(k)  and  B(k)  as  follows: 


B(k)=2 


_!  V(1-Z2)(1-/C2Z2) 

or,  since 


J%,.-2  I       -^i    (Art.  192). 
-i  VZ       JQt+iVZ 


•£• 


dz 


*  Legendre,  Fonctions  Elliptiques  (1825),  t.  I,  p.  90. 
t  Jacobi,  Werke,  Bd.  I,  p.  82  (1829). 


MODULI   OF   PERIODICITY.  213 

If  further  we  write 

z=         ~  1  =  ,  k'2=l-k2, 

Vl  -  k'2v2 

d  -k'2vdv  2_     -k'2v2  2  2  _  k'2(l  -  v2) 

^  ~  (1  -  A;'2,2)*  ~l-k'2v2  '  l-k'2v2 

it  is  seen  that 
i 

A (k)  =  2  Ck  dz        =  =2i  C1  dv       =  .       [ JacobL] 

Ji  V(l-z2)(l-A'2z2)         J0  V(l-v2)(l  -k/2v2) 

If  then  we  write 

Kf  =  r1  dv 

JQ  V(l  -  r2)(l  -  k"2v'2) 
we  have  A(k)  =  2iK'. 


The  quantity  /:'  is  called  the  complementary  modulus. 
Since  B(k)  =  4K  and  A(k)  =  2iK',  the  formulas  of  the  preceding 
article  become 

M(+  1)=-3JK:,  u(-  1)=-  K, 


x)  =  -iKr,  u(oo,-oo)^-  2K  -iK', 

,  1)=0,  w(0,-  1)  =  -  2K. 


Anticipating  what  follows,  if  we  write 

dz 


-  z2)(l  -  fc2z2) 
and  if  z  considered  as  a  function  of  M  is  written 

z  =  sn  u, 
we  have  from  the  above  formulas 

sn(-3K)=l,     sn(  -  3  K  -  iK')  =L      sn(-  iK')  =  oc  ,  etc. 

n* 

ART.  194.  We  shall  consider  next  the  moduli  of  periodicity  for  Weier- 
strass's  normal  form  of  integral  of  the  first  kind. 

We  note  that  the  point  at  infinity  is  a  branch-point  (Art.  115)  for  the 
integral 

r  dt  r  dt 

J  2v/£-e*  -  et  -  e]     J  \  5     ' 


where  S(t)  =  S  =  4(t  -  ej  (t  -  e2)  (t  -  e2). 


214  THEORY   OF   ELLIPTIC    FUNCTIONS. 

In  the  Riemann  surface  T  without  the  canals  a  and  b  let 


and  let  u(t,  Vs)  denote  the  corresponding  integral  in  T'. 


"b 

Fig.  65. 

We  may  here  write  (cf.  Art.  139) 

u(fy—  u(p}  =  A'  on  the  canal  a,  and 
^(p)  _  u(X)  =  B'  on  the  canal  b. 

The  quantities  u(ei),  u(e2},  u(e3)  may  be  computed  as  follows.     In  the 
figure  we  note  that,  when  the  integration  is  taken  in  the  upper  leaf, 

C^_dt==   PJL  +   re*_dt=  =  u(p)+u(el)-u(X)=u(el)-A'. 

J*  Vs    J*Vs    'J*  Vs 

In  the  lower  leaf  along  the  congruent  path  of  integration, 


Through  subtraction  it  follows  that* 


upper  leaf 

Similarly  along  the  upper  bank  of  the  upper  leaf  of  T  , 
_^=  p_d^+  Ce2J^  =  u(p)-u(e1)+u(e2)-u(^= 

Je,    VS       J+  VS       J*     VS 

while  for  the  congruent  path  in  the  lower  bank, 


*  Vs 

*  Cf  Riemann-Stahl,  Ellipt.  Fund.,  p.  134.  In  comparing  the  results  given  by 
different  authors  it  must  be  noted  that  in  most  cases  the  sign  of  equality  may  b^ 
replaced  by  that  of  congruence. 


MODULI   OF   PEEIODICITY.  215 

Hence  through  subtraction  it  is  seen  that  in  the  upper  leaf 

J«  VS 

We  may  therefore- write 

r*2  Hf          f*£2  rit          r*\  /it 
(II)  2  /  -£L  -  2  /   -2L  +  2  /  **=  —  A'  +  ff. 

»/oo  VS  Je,.  VS  «/oo  VS 

We  further  have  in  the  upper  leaf  of  T' ', 


V S 

=  A'  +  u(e3)-u(e2); 
while  in  the  lower  leaf 

*   dt          -x     x          -,    \ 

— =  =  u(«8)  -  tt(«j). 


Through  subtraction  we  have  in  the  upper  leaf 

9 

It  is  also  evident  that 

J«>  V/S  JK  \fi&'  Jez  V/S 

or 

an)  P4-f- 


ART.  195.    It  follows  at  once  from  (I),  (II)  and  (III)  that  in  T 

*,  __:£  —  „,    say, 

V  S  * 

z  dt        -A'  +  B' 


-,    x       C**  dt        B' 

u(e-3}=  I    —  =.=  —  =  -a>'. 

J*  VS       2 
From  these  definitions  of  w,  a>f,  to",  it  is  seen  that 

co"  =  a)  +  a/. 
Again  (cf.  Art.  185),  if  we  write  * 

-  u  = 


and  write  the  upper  limit,  considered  as  a  function  of  the  integral  u, 

t  =  p(u), 

*  The  sign  of  the  integral  is  changed  in  order  to  retain  the  notation  of  Weierstrass. 
It  is  seen  in  Chapter  XV  that  $u  is  an  even  function.     It  is  called  the  Pe-function. 


216  THEORY   OF   ELLIPTIC    FUNCTIONS. 

we  have 


(IV) 


e2 


ART.  196.     In  Art.  185  we  derived  the  relation 

Vs=    x  *vz 

If  we  write 
then  also 


where  t  =  £3  +  — 2 


dt  F   dt 

—  du  =  — =,  or    —  u  =  I     —=., 

Vs  J»  Vs. 

du    dz_}  or  JL=  r  _^L. 

Ve       VZ  Vs      Jo,iVZ 


It  follows  that 

\V  e 

and 

1 

=  e3  +  — 


It  is  also  evident  that 

K.fjfe  1     p*    =    - 

Jo.lVZ  V£  Joo   V>S        V£ 

or  w  =  VeK,  and  similarly  a/  =  -V  dK'. 

ART.  197.     T/ie  conformal  representation  of  the  T' -surf ace. 
In  Chapter  VII  we  saw  that  if 

r*>s  dz 

u  =  I         . j 

JZo>SoVR(z) 

then  z  is  a  one-valued  function  of  u.  We  also  saw  that  if  the  path  of 
integration  is  unrestricted,  more  than  one  value  of  u  correspond  to  every 
value  z,  s.  The  collectivity  of  these  values  was  expressed  by 

u  =  u(z,  s)  +  mA  4-  IB, 

where  u(z,  s)  represented  the  above  integral  in  the  simply  connected 
surface  T'  and  m  and  I  were  integers. 

If  we  write  z  =  (j>(u),  then  <j>(u)  is  a  one- valued  function  of  u.     We 

also  saw  that  s=  —  =  VR(z)  is  a  one-valued  function  of  u.     Further, 

du 

in  T'  the  quantity  u  is  uniquely  determined  as  soon  as  the  upper  limit 
z,  s  is  known.  Therefore  for  every  value  z,  s  in  the  surface  T'  we  may 


MODULI   OF   PERIODICITY.  217 

compute  the  corresponding  value  of  u  and  lay  it  off  in  the  plane  of  the 
complex  variable  u.  Since  the  integral  u  never  becomes  infinite  (Art.  136), 
it  follows  that  all  the  values  of  u  which  correspond  to  the  collectivity  of 
values  z,  s  in  the  surface  Tf  ma}'  be  laid  off  within  a  finite  portion  of  the 
u-plane. 

It  cannot  happen  that  to  two  different  values  of  2,  s  on  the  surface  T' 
there  corresponds  the  same  value  u.  For  if  this  were  possible,  then  re 
ciprocally  to  this  value  of  u  either  there  would  correspond  two  different 
values  of  z  in  the  T'-surface  and  z  would  not  be  a  one-valued  function 
of  u,  or  there  would  correspond  two  different  values  of  s,  and  then  s  would 
not  be  a  one-valued  function  of  17.  The  points  in  the  w-plane  follow  one 
another  in  a  continuous  manner  and  the  region  which  they  fill  is  simply 
covered.  It  follows  that  the  portion  of  surface  in  the  ?7-plane  and  the 
simply  connected  Riemann  surface  T'  are  conformal  representations  of 
each  other,  since  to  ever}'  point  of  the  one  structure  there  corresponds  one 
and  only  one  point  of  the  other  structure  and  vice  versa. 


u  —  Plane 

Fig.  66. 

We  may  next  investigate  the  form  of  the  portion  of  surface  in  the  77- 
plane  which  is  the  ijnage  of  the  surface  included  within  the  canals  a  and  b. 
We  compute  the  value  of  u  for  the  point  ft  which  is  the  intersection  of  the 
left  bank  of  the  canal  a  with  the  right  bank  of  the  canal  b.  The  value  of 
u  at  this  point  we  also  call  fi  and  lay  it  off  in  the  77-plane.  We  compute  for 
every  point  of  the  left  bank  of  a  the  corresponding  value  of  u  and  lay  it 
off  in  the  T7-plane.  We  thus  have  a  curve  an  in  the  77-plane  which  does 
not  cross  itself.  Let  the  other  end-point  be  denoted  by  d  in  the  77-plane, 
which  point  corresponds  to  the  point  d  on  the  surface  T'.  If  next  starting 
from  d  we  traverse  the  bank  X  of  b  and  lay  off  the  corresponding  values  in 
the  plane  77,  we  have  a  curve  b^  which  ends  at  7%  say.  Then  starting  from 
f  in  T'  we  go  along  the  bank  p  of  a  and  form  in  the  77-plane  the  corre 
sponding  curve  a p.  Finally  we  return  along  the  bank  p  of  b  back  to  /?, 
and  the  corresponding  curve  bp  in  the  T-plane  must  lead  back  to  the 


218  THEORY   OF   ELLIPTIC   FUNCTIONS. 

initial  point  /?.  The  canals  a  and  b  are  thus  conformally  represented  on 
the  tZ-plane. 

Since  the  canals  a  and  b  are  the  boundaries  of  T' ,  the  curve  a^b^apbp 
must  bound  the  surface  which  is  the  conformal  representation  of  T'  in 
the  w-plane.  The  interior  of  the  figure  is  this  conformal  representation, 
for  u  cannot  be  infinite  for  any  value  of  z,  s,  which  may  be  the  case  if  the 
surface  without  the  curve  aA6Aap6p  represented  conformally  T' '. 

Remark.  —  The  curves  ax  and  ap  are  parallel  curves,  that  is,  to  every 
point  on  a  A  there  corresponds  a  point  on  ap,  so  that  lines  joining  such 
pairs  of  points  are  equal  and  parallel.  For  if  we  take  on  the  canal  a  in 
T'  two  points  opposite  each  other  on  the  left  and  the  right  banks  respec 
tively,  then  we  have 

u(X}  -  u(p)  =  A. 

Consequently  the  complex  quantity  A  represents  the  length  and  the 
direction  in  the  w-plane  of  the  distance  between  two  points  lying  on 
opposite  banks  of  the  canal  a,  which  conformally  in  the  w-plane  lie  on  the 
curves  a*  and  ap.  Since  A  is  a  constant  the  two  curves  a^  and  ap  must 
be  parallel. 

Similarly  b^  and  bp  are  parallel  curves  and  the  distance  between  them 
is  B. 

If  the  variable  crosses  a  canal  a  or  b  in  T' ',  we  have  values  of  u  which 
lie  in  a  period-parallelogram  that  is  congruent  to  the  first  parallelogram, 
and  by  crossing  the  canals  a  and  b  arbitrarily  often  in  either  direction  we 
have  more  and  more  parallelograms  which  completely  fill  out  the  w-plane. 

ART.  198.  The  form  of  the  two  canals  a  and  b  was  arbitrary.  We 
shall  show  that  they  may  be  taken  so  that  the  corresponding  parallel 
ogram  in  the  w-plane  is  straight-lined.  As  a  somewhat  special  case  take 
Legendre's  normal  form  and  let  the  modulus  k  be  real,  positive  and  less 
than  unity. 


-i/k 


C+i)  +K,' 


Fig.  67. 


We  draw  the  canals  a  and  b  so  that  they  lie  indefinitely  near  the  real 

axis  and  indefinitely  close  to  the  points  —  1,  +  1,  +  -,  as  shown  in  the 
figure.  k 


We  had  in  T' 


-  =  rzs  dz 

t/o,i  VZ 


The  differential  dz  is  here  real,  being  taken  along  the  right  and  left 
banks  of  the  canals  which  are  supposed  to  lie  indefinitely  near  the  real  axis. 


MODULI   OF   PEEIODICITY. 


219 


For  the  bank  /  of  a  we  have  1  —  z2  >  0  for  all  points  except  z  =  1,  or 
z  =  —  1,  and  consequently  also 

1  -  Fz2  >  0. 

It  follows  that  a  A  is  real  in  the  "w-plane,  since  u  is  real  for  all  points  on  the 
bank  X  of  a.  Hence  in  the  TT-plane  a^  coincides  with  the  real  axis. 

On  the  bank  X  of  b  we  have 

1  -  z2  <  0    and    1  -  k'2z2  >  0. 

The  elements  of  integration  are  therefore  all  pure  imaginaries  along  this 
bank  and  consequently  u  is  purely  imaginary  along  this  bank.  It  follows 
that  bji  in  the  w-plane  is  a  straight  line  that  stands  perpendicular  to  the 
axis  of  the  real. 

Since  ap  is  parallel  to  a  A  and 
bp  to  b  A,  the  conformal  repre 
sentation  of  T'  on  the  w-plane 
is  a  rectangle  with  the  sides  a* 


dr  =  A(k)=2iK', 

We  may  represent  the  inte 
gral    in   Weierstrass's  normal 
form  conformally  in  a  like  manner, 
student. 

As  another  exercise  derive  the  results  of  this  Chapter  by  taking  the 
Riemann  surface  as  indicated  in  Fig.  68. 


This  is  left  as  an  exercise  for  the 


1.  Show  that 

2.  Show  that 

3.  Prove  that 

4.  The  substitution 
transforms 


EXAMPLES 


/»<»,- oo   fJ7 

/  -2==  =  K 

Ji         VZ 


</  = 


0)   = 


V40-  ej  (*-  e2)  (t- 


into 


r 

J  €•> 


ds 


V  4  (s  -  ej  (s  -  e2)  (s  - 


How  does  this  result  compare  with  the  one  derived  by  the  methods  of  this  Chapter? 
5.   Derive  by  means  of  the  Riemann  surface  the  formula 

dt          Cei  dt 
— =+/      —;= 

+    VS        J^   VS 


CHAPTER  X 
THE  JACOBI  THETA-FUNCTIONS 

ARTICLE  199.     We  saw  in  Chapter  V  that  the  4>-f unctions  of  the  second 
degree  satisfied  the  two  functional  equations 

$(u  +  a)=  &(u), 


If  Q  =  e    a,  we  saw  in  Art.  87  that 

m  =  +  oo  4  id 


mu 


We  have  now  to  write :          "&  =  B  (k)  =  4 


It  follows  that  Q  =  e    2  K . 
If  further  we  write 

we  have  * 

m  —  +  oo  Tti 

-==  mu 


m=—  oo 
TO  =  +oo 


When  K  and  K'  are  introduced  into  the  functional  equations,  they  are 


=  e~*(U     ^ 

In  Oi(w)  the  term  which  corresponds  to  m  =  0  is  unity.     If  we  take 
this  term  without  the  summation  and  then  combine  under  the  summa- 

*  Cf.   Jacobi,  Werke,  Bd.  I,  pp.  224  et  seq.;  and  in  particular  Hermite,  Cours 
redige  en  1882  par  M.  Andoyer,  p.  235  (Quatrieme  Edition). 

220 


THE   JACOBI   THETA-FUNCTIONS.  221 

tion  the  term  which  corresponds  to  +  m  with  the  term  that  corresponds 
to  —m,  we  have 


5)  <r2U  * 


TTlfflM 

~K~ 


m=l 


or 

HZ  =  1 

The  terms  in  HI(W)  may  be  combined  as  follows: 


l 

l, 


It  follows  immediately,  as  we  have  already  seen  in  Art.  148,  that 


ART.  200.     \Ye  introduce  two  new  functions,*  the  first  of  which  is 
defined  by  the  relation 

H(u)=Hi(M  -K). 
We  have  at  once 

771=30       /2m+l\2 


H(»)  =  2  X  ,~  cos        "          ™  -  (2  m  +  1) 

771  =0 

But,  since  cos  Li  -  (2  m  +  !)-=(-  1)TO  sin  A,  it  is  seen  that 


m=x 


«  =  2         (  -  l-  sin 

0 


The  second  function  is  defined  by  the  relation 

e(«)-'ei(«-/io 

and  consequently  we  have 


m=  1 

It  is  seen  at  once  that 

0(-    M)=0(M). 


The  functions  0i(w),  HI(W),  0(^),  H(M)  are  known  in  mathematics  as 
the  0-functions.  Excepting  H(w)  they  are  all  even  functions,  and  it  is 
seen  that  they  are  more  rapidly  convergent  than  a  geometrical  progression. 

*  a.  Jacobi,  loc.  tit.,  p.  235,  and  Werke,  II,  p.  293;  see  also  Hermite,  loc.  tit.,  p.  235. 


222  THEOEY   OF  ELLIPTIC   FUNCTIONS. 

ART.  201.     From  the  equation 


(U  +  K)  =  H(u  + 
M  +  K)  =  -  H(w). 


0i 


it  follows  that 

and  therefore 

In  a  similar  manner 

We  also  have 

9(* 

and  0(tt 

We  thus  have  the  four  formulae 


(I) 


From  these  formulae  we  derive  at  once 


(ID 


From  (I)  and  (II)  we  again  have 


2  K  = 


and  finally 


(III) 


(IV) 


0(ti 


ART.  202.     We  shall  next  increase  the  argument  of  the  0-functions 
by  iK'. 
We  have 


m=  -oo 

OT=  +00 


m=  +00 


*-  X 


m  =  -oo 
m=+oo 


(TT)'-I. 


1        Jrt'tt 


m=-oo 


THE   JACOBI   THETA-FUNCTIONa 

If  further  we  write  * 


JIT  IU 


we  have 


0i  (z*  +  iK')  =  ^(M)Hi(u). 


We  may  also  note  that 


e 


2m+l\  2 

-     1 


le        2K         e  2K 


m=+x     /2m+l\2     2m+l    (2m  + 

— »  (    — I     +  — JT-; 


1    (m+l)rt« 


'e     Ke*K    2K, 


=  2) 


where  m  +  I  =  mf. 
It  is  seen  at  once  that 

Since 

we  have  00 


—  -  —  (u-K) 
4K     2K(       W 


In  a  similar  manner  it  is  seen  that 
We  may  therefore  write 

(V) 

It  follows  from  (I)  and  (V)  that 
(VI) 


HI(M  +  K  +  iK')  =  -il( 
0i(u  +  K  +  i,K')  -    U(u 


*  Hermite,  loc.  cit.,  p.  236. 


223 


224  THEORY   OF  ELLIPTIC   FUNCTIONS. 

It  is  clear  that 

HI(M  +  2iK')  =  H,[(u  +  iK')  +  iK'] 


If  we  put 

it  follows  that 

We  have  the  following  formulae : 

(VII)       ®i(^  +  2iKf) 


It  is  seen  that  H  and  0  satisfy  the  functional  equations 
&(u  +  4K)  =  $(u),     3>(u  +  2  t/g;')  =  -^(u) 
while  HI  and  @j  satisfy 


We  note  in  particular  that  the  four  theta-functions  belong  to  two 
categories  of  functions  of  essentially  different  nature. 

ART.  203.  The  Zeros.  —  The  ©-functions  being  $-f  unctions  of  the  second 
degree  vanish  at  two  incongruent  points  (congruent  points  being  those 
which  differ  from  one  another  by  multiples  of  4  K  and  2  iK'). 

We  saw  in  Art.  200  that  H(w)-  was  an  odd  function  and  therefore  vanishes 
for  u  =  0.  We  also  had 


and  consequently 

H(2K)  =  -H(0)  =0. 

The  points  0,  2  K  are  therefore  the  two  incongruent  zeros  of  this  func 
tion;  i.e.,  the  function  H(V)  vanishes  on  all  points  of  the  form 


2K  +  m24K  +l22iK', 
where  mi,m2,li,l2  are  integers. 

Hence  all  the  points  at  which  H(w)  vanishes  are  had  for  the  values  of 
the  argument 

u  =  m2K  +  n2  iK' , 

where  m  and  n  are  integers. 


THE   JACOBI   THETA-FUNCTIONS.  226 

Further,  since 

Q(u  +  iK')=M(u)H(u), 
when  u  =  0,  we  see  that 

and  since 

e( 
we  also  have 


iK'  +  2K)=Q. 
The  zeros  of  0(^)  are  consequently 

m2K  +  (2n  +  \)i 
By  definition  we  have 

fi(M)-Hi(w~i 

so  that  the  zeros  of  HI(W)  are 

(2m  +  1)K  + 
Finallv,  since 

0(M)  =  0i(u- 

the  zeros  of  0i(w)  are 

(2m  +  1)#  +(2rc  -f 
ART.  204.     Write 


where  q  =  e     K  ; 

and 


m=-°o 

where  qQ  =  e 

It  is  seen  that  the  latter  series  fulfills  the  requirements  of  convergence 
given  in  Art.  86. 

We  also  note,  cf.  formulas  (II)  and  (VII),  that 

0 1  ( u •  K',  iK)  and  e ~  TKK'Ql (iu;  K,  iK') 
satisfy  the  same  functional  equations 


The  two  functions  have  also  the  same  zeros 
u  =(2m  +  l)K'  +  (2n 
It  follows  that  the  ratio  of  the  two  functions  is  a  constant. 


226  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  therefore  have  (cf.  Jacobi's  Werke,  Bd.  I,  p.  214) 


e    4KK'®i(iu;  K,  iK')  =  C8i(t*;  Kl ',  iK), 


/Hi(ra;  K,  iK')  =  C®(u;  K',  iK), 


e    *KK"H.(iu;  K,  iK')  =  iCH(u;  K',  iK)} 


e    *KK'®(iu;K,iK')  = 


EXPRESSION  OF  THE  THETA-FUNCTIONS  IN  THE  FORM  OF 
INFINITE  PRODUCTS. 

ART.  205.     With  Hermite  *  consider  the  two  functions 
$(tO  =  <f>(u  +  iK')  </>(u  +  3i 


and 


It  is  seen  at  once  that  if  (f>(u)  has  the  period  2  K,  then 


It  is  also  evident  that 


and  consequently 


If  next  we  put 

niu 

(j>(u)  =  1  +  e  K  , 
we  have 


and  also 

<>-  u 


. 
K. 

*  See  ./Vote  swr  la  theorie  des  f  auctions  elliptiques  placed  at  the  end  of  Serret's  Calcid 
Differentiel  et  Integral,  pp.  753  et  seq.;  CEuvres,  t.  2,  pp.  123  et  seq. 


THE   JACOBI   THETA-FUNCTIONS.  227 

It  is  thus  seen  that 


(n  =  1,  3,  5,  .  .  .  ) 
and 


C7TtU\ 
1  +  e*]II(l  +  2?n 

(n  =  2,  4,  6,   •  •  •  .) 


These  products  are  convergent  (cf.  Art.  17)  if  |  q  \  <  1  (see  Art.  81). 

ART.  206.     The  two  functions  &(u),  <&i(u)  both  have  the  period  2  K 
and  they  satisfy  the  functional  equations 


Let  us  introduce  a  function  W(u)  denned  by  the  equation 


We  have  at  once 

si  /        iK' 


$(M),     ^(H  +  2  K)  =  -  V(u). 
It  is  evident  from  formulas  (II)  and  (V)  that  we  may  write  (cf.  Art.  83) 


where  A  is  a  constant. 
Noting  also  that 


it  is  seen  that 


=  2  A  t/sinul-22cos2u  +    *}l-2*  cos2  u 


where  A  is  a  constant. 


228  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  207.     To  determine  the  constant  A  of  the  preceding  article,  we 
follow  a  method  due  to  Biehler.* 

Consider  the  product  composed  of  a  finite  number  of  factors 

(1)  f(t)  =  (1  +  qt)  (1  +  230   .   .   .    (1  +  q2»-ty 


This  expression  developed  according  to  positive  and  negative  powers  of 
t  is  of  the  form 

(2) 

The  following  identity,  which  may  be  at  once  verified, 
f(q2t)  (q2n  +  qt)  =  f(t)  (I  +  q2n+lt), 
gives  between  two  consecutive  coefficients  Ai  and  AI-I  the  relation 


We  thus  have 


4.       -4. 


1    - 


If  these  equations  are  multiplied  together,  we  find  that 
A        A  ^     (1  -g^H1  n- 


But  it  follows  directly  from  (1)  and  (2)  that 

An  =  qn*. 
We  therefore  have 

A°  =          (1  -  q2}  (1  -  g4)  ...   (1  -  92  ») 

When  n  becomes  indefinitely  large,  it  is  seen  that 

\ 

°  ~~  (1  -  q2)  (1  -  q4)  (1  -  q6)   .   .   .' 

*  Biehler,  Crelle,  Bd.  88,  pp.  185-204;  see  also  Hermite,  loc.  tit.,  pp.  770-772;  Appell 
et  Lacour,  Fonctions  Elliptiques,  pp.  398-399.  Jacobi  gives  two  methods  of  deter 
mining  this  constant  (Werke,  I,  p.  230,  §  63  and  §  64)  and  a  third  proof  (Werke,  II, 
pp.  153,  160). 


THE   JACOBI   THETA-FUNCTIONS.  229 

Further,  since  A. 

it  follows  from  the  equation  (2)  that 

(1  +  qt)(l  +  <?30(1  +  550   .   .   .     l  +         1  4- 


(1  -  g2)  (1  -  q*)  (1  -  g6)   ... 
Writing  t  =  e2iu,  this  formula  becomes 

(1  +  2  q  cos  2  u  +  q2)  (1  +  2  g3  cos  2  u  +  q6)  -  •  • 

=  1  -f  2  q  cos  2  H  .  -i-  2  g4  cos  4  K  -  •   •   • 

(1    -  92)   (1    -  54)   (1    -  Q6)     ... 

From  this  we  conclude  that  the  constant  A  of  the  previous  Article  is 

A  =  (i  -  92)  (i  -  g4)  (i  -  ?6)  •  •  •  ; 

and  at  the  same  time  it  is  shown  that  0!  as  denned  in  the  last  Article  as 
an  infinite  product  is 

Q   f2J±!L\  =  i  +  2  q  cos  2  u  +  2  g4  cos  4  u  +  '  •  •  , 

or  ®i(«)= 

which  is  the  original  definition  of  this  ©-function. 

Example.  —  By   means   of   the   infinite   products  prove   the   formulas 
(I),  (II),  (III),  (IV)  and  (V)  of  this  Chapter,  and  therefrom  derive  the 

.    .  i  .,         .        f  TT   /2Ku\    u/2Ku\       ,r./2Ku\ 
expressions  in  infinite  series  of  HI  I  —  I  H(—  —  —  1  and  0f  -  J- 

THE  SMALL  THETA-FUNCTIOXS. 

ART.  208.     Jacobi  (Werke,  Bd.  I,  pp.  499  et  seq.)  introduced  a  notation 
similar  to  the  following  (see  Art.  210): 

m=  -»-Qo 

0(2  Ku)  =  &Q(U)  =    5     (~  l)mgmV2  """'", 


m=  - 


TO=  —  ao 

m=  4-ao    /27n.-t-lv- 

- 


—  30 

-1-30 


230  THEORY   OF   ELLIPTIC   FUNCTIONS. 

It  follows  at  once  [cf.  formulas  (I)  and  (V)]  that 

(T) 


and  if  T  =        , 


(V) 


J  r)  = 
i  T)  = 


The  other  formulas  given  in  the  Table  of  Formulas,  No.  XXXIII,  are 
left  as  examples  to  be  worked. 

ART.  209.     For  brevity  we  may  write 

m  =  <x>  wi  =  oo 

e<>  =  n  a  -  22m)>    QI  =  n  (!  + 


m  =  l 

~m  =  <x> 


m  =  l 

m  =  oo 


It  follows  at  once  that 


-  2  g2™-i  cos  2  ;m  +  q4m~ 

l 

1  m  =  oo 

4sin7rw  JJ  (1  -  2q2mcos2xu 

m  =  l 

1  TO  =  00 

4  cos  TTW  JJ  (1  +  2^2w  cos  2  TTU 


If  we  write  2  =  eiM7r,  we  have 

COS  2  7m  + 


_  q2m+lz-2      I  _ 


sin „ /?m.±I r  _  M\       sin „ /i^L+j  t  +  tt\ 

=  \  ^ /  »-iun  \  2 / 


sn 


.    2  m  +  1 

sm TTT 


gittTT 


THE   JACOBI   THETA-FUNCTIONS. 

• 

We  therefore  have 


n/i 

m=1l 


m_ 

772  =  00 


sin2  (m-r 
sin2  7:11    \ 


cos^ 


(mm)/ 


sm-  TT 


-w- 


ART.  210.  Jacobi's  fundamental  theorem.  If  we  write  r.u  =  x  on  the 
right-hand  side  of  the  equations  above,  the  theta-functions  as  given  by 
Jacobi*are  m  =+x 


m=  -x 

=  1  -  2  g  cos  2  x+  2  g4  cos  4  x  —  2  $9  cos  6  x  + 

771  =  +00 

fe  q)=  i2(-i) 


sin  z  —  2  ^t  sin  3  x  +  2  q"  sin  5  x  —   •  •  •  , 


=  2  g*  cos  z  +  2  ^  cos  3  x  +  2  q3*  cos  5  re  +  2  3*  cos  7  x  + 

m=  +  oo 

#3(*,  9)  =    5)  ^^e2""* 

m=  —  oo 

=  1  +  2  q  cos  2  x  +  2  g4  cos  4  x  +  2  g9  cos  6  x  +    •  •   •  . 
We  have  at  once 


+  J  log  5  •  0  =  -iq-*exi$i(x) 
+  l  log  g  •  0  =  -iq-*exi$0(x) 
+  I  log  q  •  0  -  g-V%Or) 
+  i  log  q  •  i)  =  q- 


#0(x  +  log  9  •  i)  =  -  g-V^' 


+  log  9  •  i)  =q- 
+  log  5  •  i)  =  g-^ 


+  ^  -  +  J  log  g  •  0  = 


&2(x  +  i  TT  +  i  log  9  •  i)  =  iq-*exi$o(x) 
&3(x  +  \-  +  Hogg  •  0  =  -  ^-V^i( 

*  Jacobi,  Werke,  I,  pp.  497-538. 


232 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


We  next  observe  that  if  the  quantities  a,  6,  c,  d;  a',  b',  c',  d'  are  con 
nected  by  the  equations 

a'  =  %(a  +  b  +  c  +  d), 
b'  =  %  (a  +  b  -  c  -  d), 
c'  =  i(o  -  b  +  c  -  d}, 
d'  =  0  -  6  -  c  +  d 


(D 


it  follows  that 


(2) 


c  =  i(a'-  b'+  c'-  d'}, 

and  also  that 

(3)     a2  +  b2  +  c2  +  d2  =  a/2  +  b'2  +  c'2  +  d'2. 

We  shall  next  show  that  if  a',  ~bf,  c'y  d'  are  either  all  even  integers  or 
all  odd  integers,  then  also  a,  b,  c,  d  are  all  either  even  or  odd  integers. 

This  may  be  seen  at  once  from  the  following  table.* 

We  note  that  all  integers,  positive  or  negative,  belong  to  one  or  the  other 
of  the  four  forms 

where  p  is  an  integer  or  zero. 

For  four  even  integers  we  may  write 

a  =  b  =  c  =  d  = 


4/9 
4/9 

4/? 

4/9 


4r 

4r 


4^ 
4(5 
4^ 
4^ 


where  the  numbers  in  any  column  may  be  permuted  among  one  another. 
If  for  brevity  we  put 


a  —  ft  +  Y  —  d  =  f 
it  follows  from  equations  (1)  that 
a'-  6'= 


2  a' 

2a>+  I 
2a'+2 
2a;+3 


2p' 


2r'-l 


2d' 

2^+1 
2  d' 

2d'-l 
2  d' 


*  See  Enneper,  Elliptische  Funktionen,  p.  136. 


THE   JACOBI   THETA-FUNCTIONS.  233 

For  four  odd  integers  we  may  write 

a  =  b  =  c  =  d  = 

4a  +  1  4/?+  1  4r  +  1  4d  +  1 

4a  +  1  4/?+  1  4r  +  1  40+3 

4a  +  1  4/?  +  1  4r  +  3  4£  +  3 

4a  +  1  4/9  +  3  4r  +  3  4  d  +  3 

4a  +  3  4,3  +  3  4r  +  3  40  +  3 

where  the  corresponding  values  of  a',  b',  c',  d'  are,  owing  to  equations  (1), 
a'  =  6'=  c'  =  d'  = 

2a'+2 
2a'+3 

2a'+4  2/3'-  2  2  r' 

2a'+  5 
2a'+6 
If  for  example  we  write 

a  =  I,     b  =       3,     c  =       5,     d  =  7, 

we  have  a'=  8,     b'  =  -  4,     c'  =  -  2,     dr  =  0; 

and  reciprocally  if          a  =  8,     b  =  —  4,     c  =  -  2,     d  =  0, 
we  have  a'=  1,     £/=       3,     cr=       5,     d'=  7. 

It  follows  that  if  for  a,  6,  c,  d  we  write  all  possible  combinations  includ 
ing  all  systems  of  four  even  integers  and  all  systems  of  four  odd  inte 
gers,  the  corresponding  integers  a',  b'  ,  c',  d'  will  take  the  same  systems  of 
values  in  a  different  order  and  in  such  a  way  that  none  of  the  systems 
will  be  omitted  or  doubled. 

Since  x2  l 

&3(X)   =   ^q^e2m,i=^em^ogq  +  2mxi  =  ^^^^ 

and  ^2W 

it  follows  that  a    (u2+x2+yt  +  z2)         L 


1  \T 

—  (w2  +  x2  +  v2  +  z2 

and  ' 


where 

L  =  (2  v  *  Iog9  +  u^')2  +  (2  v' 

+  (2  v"  J  log  ?  +  i/i)2+  (2  i/"  J 

and  M 


ev" 


the  summation  in  the  first  equation  to  be  taken  over  all  positive  and 
negative  even  integers  2  v,  2  !/,  2  v",  2  i/"  and  in  the  second  equation  over 
all  positive  and  negative  odd  integers  2  v  +  1,  2  i/+  1,  2  i/'+  1,  2  r/"+  1. 


234  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Adding  the  two  expressions  we  have 

—  —      2  +  xz  +  vz  +  2 

(4)   ^3 
where 


the  summation  to  be  taken  over  all  systems  of  four  even  integers  a,  b,  c,  d 
plus  the  summation  over  all  systems  of  four  odd  integers  a,  6,  c,  d.  . 
We  note  that  N  may  be  written  in  the  form 

(5)     jv  =\a  +  b  +  c  +  d  lQg?  +  w  +  x  +  y  +  z  H2 

L            2  2  2               J 

fa  +  b  —  c  —  d  log  q  ,   w  4-  x  —  y  —  z  .I2 

|_            2  2  2           "  J 

fa  —  b  +  c  —  d  log  q  .   w  —  x  +  y  —  z  .~j2 

L            2  2  2             J 

.  [a  -  6  -  c  +  d  log  g  ,   w  —  x  —  y  +  z  -I2 

[2  2  2               J  ' 

We  define  w',  xf,  yf,  z'  through  the  equations 

'  =  %(w  +  x  +  y  +  z),     y'  =  $(w  -  x  +  y  -  z), 


/  s*\ 

J  x'  =  %(w  +  x  —  y  —  z),     z'  =%(w  —  x  —  y  +  z). 
It  follows  at  once  that 

w'2  +  x'2  +  y'2  +  z'2-  =w2  +  x2  +  y2  +  z2. 

If  further  we  put  accents  on  all  the  letters  in  equation  (4)  and  note 
that  the  summation  taken  over  all  systems  of  four  even  integers  a',  b',  c',  d' 
plus  the  summation  over  all  systems  of  odd  integers  a',  b',  c',  d'  is  in 
virtue  of  (1)  and  (5)  the  same  as  those  above  over  a,  b,  c,  d,  it  follows  that 


Jacobi  (loc.  cit.)    made   this  formula   the   foundation  of   the   theory  of 
elliptic  functions. 

ART.  211.     If  for  if  we  write  w  +  n,  we  have 


while  at  the  same  time  w'  ,  -x',  y',  z'  are  increased  by  J  n  so  that  $3(w'  + 
becomes  #0(^1)  and$2(X+  i)  becomes  —  &i(w'). 
The  formula  above  becomes 


The  number  of  formulas  which  we  may  derive  in  this  manner  is  thirty- 
five,  which  fall  into  two  categories,  namely,  changes  in  w,  x,  y,  z  which 


THE   JACOBI   THETA-FUNCTIOXS.  .        235 

produce  corresponding  changes  of  £  n  and  £  log  q  •  i  in  w'  xf,  y',  z'  and 
secondly  changes  in  w,  x,  y,  z  which  cause  changes  of  J  n  and  \  log  q  •  i 
in  w',  xf,  y',  z' . 

The  following  eleven  formulas  belong  to  the  first  category,  where  for 
brevity  we  write 

()^.p}  for 

and  (Ipvp)'  for 

(A). 

(1)  (3333)  -f  (2222)  =  (3333)'  +  (2222)' 

(2)  (3333)  -  (2222)  =  (0000)'  +  (1111)' 

(3)  (0000)  +  (1111)  =  (3333)'-  (2222)' 

(4)  (0000) -(11 11)  =  (0000)' -(11 11)' 


(5)  (0033)  +  (1122)  =  (0033)'  +  (1122)' 

(6)  (0033)  -  (1122)  =  (3300)'  +  (2211)' 

(7)  (0022)  +  (1133)  =  (0022)'  +  (1133)' 

(8)  (0022)  -  (1133)  =  (2200)'  +  (3311)' 

(9)  (3322)  +  (0011)  =  (3322)'  +  (0011)' 
(10)  (3322)  -  (0011)  =  (2233)'  +  (1100)' 


(11)  (3201) +(2310)  =(1023)' -(0132)' 

(12)  (3201)  -  (2310)  =  (3201)'  -  (2310)' 

Equations  (11)  and  (12)  are  counted  as  one  equation,  since  (11)  becomes 
(12)  when  x,  w,  z,  y  are  written  for  wi  x,  y,  z. 
We  also  note  that  the  equations 

(5)  (7)  (9)  (11)   are  transformed  into 

(6)  (8)  (10)  (12)  and  vice  versa, 

when  —  x,  —  y  are  written  for  x,  y,  and  consequently  also  w'  becomes  zf 
and  x'  becomes  y'. 

If  we  put  w  =  x  +  y  +  2, 

it  follows  that         w' =  x  +  y  +  z,     x'  =  x,     y'=  y,     z'  =  z; 

while  if  we  write  w  =  —  (x  +  y  +  z) , 

we  have         w'=0,     x'=-(y  +  z),     \f=-(x  +  z),     z'=-(x  +  y). 

Equations  (A)  may  then  be  combined  into  double  equations.  If  for 
brevity  we  denote  $0(0)#>iO/  +  z)$ft(x  +  z)&v(x  +  y)  by  |  Qtfjiv  |  and 
&x(x  +  y  +  z)^ft(x)^v(y)^p(z)  by{^y(0},  the  five  most  interesting  of  these 
double  formulas  are  given  in  the  following  table. 


236 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


(B). 

|  0000  -  {3333}  -  {2222}  =  {0000}  +  {llll } 
|  0033  |  =  {0033}  -  {1122}  -  {3300}  +  {2211 } 
|  0022  |  -  {0022}  -  { 1133}-  {2200}  +  {3311 } 
|  0011  |  =  {3322}  -  {2233}  =  {0011 }  +  { 1100 } 
|  0123  =  {3210}  +  {2301 }  -  { 1032}  -  {0123} 

We  may  derive  a  more  special  system  of  formulas  if  in  the 

in  table  (A)  we  put 

w  =  x,         y  =  z, 

w'=x  +  y,        x'=x-yj         ?/=0,         z'=0; 

or  if  we  put 

w  =  —  x,         y  =  -  z, 

ti>'=0,         z'=0,         y'=-(x-y),         z' =  -  (x  + 
Similar  formulas,  making  in  all  thirty-six,  are  had  by  writing 

w=  y,  x=  z;  w' 
w=-y,  x=-z-,  w' 
w=  Zj  x=  y;  w' 

Using  the  notations* 


formulas 


y). 


these  thirty-six  formulas  are  included  in  the  following  table. 

(C). 

(1)  [3333]  -  (3333)  +  (1111)  =  (0000)  +  (2222) 

(2)  [3300]  -  (0033)  +  (2211)  -  (3300)  +  (1122) 

(3)  [3322]  -  (2233)  -  (0011)  -  (3322)  -  (1100) 

(4)  [3311]  =  (1133)  -  (3311)  =  (0022)  -  (2200) 


(5) 
(6) 
(7) 
(8) 


[0033]  -  (0033)  -  (1122)  -  (3300)  -  (2211) 
[0000]  =  (3333)  -  (2222)  =  (0000)  -  (1111) 
[0022]  =  (0022)  -  (1133)  =  (2200)  -  (3311) 
[0011]  -  (3322)  -  (2233)  =  (1100)  -  (0011) 
*  Koenigsberger,  Elliptische  Functionen,  p.  379. 


THE   JACOBI    THETA-FUNCTIONS.  237 

(9)  [2233]  -  (3322)  +  (0011)  =  (2233)  +  (1100) 

(10)  [2200]  =  (0022)  +  (3311)  =  (1133)  +  (2200) 

(11)  [2222]  =  (2222)  -  (1111)  -  (3333)  -  (0000) 

(12)  [2211]  =  (1122)  -  (2211)  =  (0033)  -  (3300) 


(13)  [0202]  =  (0202)  +  (1313);      [0220]   =  (0202)  -  (1313) 

(14)  [3232]  =  (3232)  +  (0101);      [3223]   =  (3232)  -  (0101) 

(15)  [0303]  =  (0303)  +  (1212);      [0330]   =  (0303)  -  (1212) 


(16)  [0213]  =  (1302)  +  (0213);      [0231]   =  (1302)  -  (0213) 

(17)  [3210]  =  (0132)  +  (3201);      [3201]   -  (0132)  -  (3201) 

(18)  [0312]  =  (1203)  +  (0312);      [0321]   =  (1203)  -  (0312) 

If  in  the  above  formulas  we  put  x  =  y,  we  have  from  (1),  (2)  and  (11) 
the  following: 

#3^3(2  x)  =  tV(*)  +  #i4 (*)  -  tVM  +  <VM 


If  we  write  y  =  0  in  the  formulas  (C),  (1),  (2)  and  (11),  we  have  the 
formulas  of  the  following  table. 

(D). 

(i)  tVWW  -  tfoWM  +  t^WC*) 

d')  #32<V«    =  tV#32(*)    +  TWO*) 

(2)  #3WW    -  #22#32(*)    -  tVVW 

(3)  ^2^12(^)    -  tWC*)    ~  ^0^22(X) 

If  in  equation  (1)  we  put  x  =  0,  we  have 

<v  -  <v  +  ^, 

or 

[1  +  2q  +  2g4+  2qg+    •  •   •   ]4=  [1  -  2  q  +  2  ?4-  2  qg+    -  -  -  ]4 
+  16g[l  +g1-2+g2'3+  q3A  +    -  •   •   ]4. 

ART.  212.  We  have  denned  and  developed  the  theta-f unctions  by 
means  of  infinite  power  series.  These  functions  being  integral  transcend 
ents  are  susceptible  of  the  treatment  indicated  hi  Chapter  I  and  per 
formed  there  for  sin  u. 

It  will  be  shown  later  (Chapter  XIV)  that  these  theta-functions  are  to 
a  constant  factor  the  same  as  the  Weierstrassian  sigma-f unctions. 

In  order  to  observe  the  general  theory  from  another  point  of  view  and 
at  the  same  time  study  Weierstrass's  presentation  of  the  subject,  we  shall 
develop  the  sigma-functions  by  means  of  infinite  binomial  products  as  has 
been  suggested  in  Chapter  I  for  sin  u.  It  is  therefore  superfluous  here 
to  express  the  theta-functions  through  these  infinite  binomial  products. 


238  THEORY   OF   ELLIPTIC    FUNCTIONS. 

EXAMPLES 
1.    Show  that 


4KK' 


[Jacobi,  Werke  I,  p.  226.] 

2.  Derive  the  corresponding  formulas  for  @t  and  H^ 

3.  If 

K'  K 

~*7T  ~*~K' 

q  =  e       K,q0  =  e       K, 

so  that  q,  q0  are  interchanged  when  K,  K'  change  places,  and  if 

@(w,  q)  =  1  -  2  5  cos  2  w  +  2  g4  cos  4  w  -  2  #9  cos  6  w  +  •  •  •  , 
H(w,  q)  =  2  ^g"sin  u  -  2  ^9  sin  3  w  +  2  "N/g^sin  5  w  -  •  •   •  , 

prove  that 


[Jacobi,  Werke,  I,  p.  264.] 
4.   Using  the  Jacobi  notation  show  that 

#0(u  +  mi  log  g)  =(-l)w<?-"l2e2mMi^0(^), 


i?2(M  +  mi  log  g)  =  q- 
$s(u  +  milogq)  -  q- 

5.   Show  that,  if  n  and  m  are  integers, 


2m          ^  1()g       =  Q^      ^(ns  +  mi  log  9)  = 


CHAPTER   XI 
THE  FUNCTIONS  snu,  cnu,  dnu 

ARTICLE  213.  It  was  shown  in  Art.  152  that  z  may  be  expressed  as 
the  quotient  of  two  ^-functions  in  the  form 

z  = 
where  u  =  I* ' 

*J ZQ,S, 

If  we  put 

u=l 

V(l-z2)(l-/;2z2) 

and  study  a  quotient  of  4>-f unctions,  it  is  seen  that       ^  must  =  0,  for 

<&i(u) 
z  =  0  in  both  the  upper  and  the  lower  leaves  of  the  Riemann  surface; 

and  further  for  z  =  ao,  we  must  have       ^u'  =oc    in  both  leaves.      It 
follows  that  ®i(u) 

^   =  0  for  z  =  0,  s  =  +  1  and  for  z  =  0,  s  =  —  1; 

$fu) 

and  =QC  for  z  =  x ,  s  =  -f-  oc  and  for  z  =  oc ,  s  =  —  oo. 

In  Art.  193  we  saw  that 

77(0,  +  1)  =  0,     77(0,  -  1)=-2K; 
and  consequently 

H[w(0,+  1)]  =  0,     H[w(0,-  1)]  =  H(-  2A')=0. 

Hence  it  is  shown  that  H(w)  becomes  zero  for  2  =  0.  s  =  + 1  and 
for  2=0,  s  =  —  1.  We  may  therefore  take  H(w)  as  the  numerator  in 
the  quotient  of  ^-functions. 

On  the  other  hand  we  have 


and  since 

0(-  iK')  =  0,    0(-  2  K  -  iK')  =  0, 

we  may  use  0  (u)  as  the  denominator  of  the  above  quotient.     If  then  for 
u  we  write  Legendre's  normal  integral  of  the  first  kind,  it  is  evident  that 

239 


240  THEORY   OF   ELLIPTIC    FUNCTIONS. 

the  quotient  3-^  has  the  desired   zeros  and  infinities,  and  has  besides 

9(tt) 

no  other  such  points. 
It  follows  that 


where  C  is  a  constant. 

To  determine  the  constant  C,  write  z  =  1  and  we  have 


= 

But  since  (Art.  193)  u(l)  =  -3  K,  we  have 

rH(-3g) 

°0(-3K)* 
In  Art.  201  we  saw  that 

Hi(N  +  3£)=H(u), 
or  H1(0)=H(-3K). 

In  a  similar  manner  it  may  be  shown  that 

0i(0)  =  e(-3K). 
We  thus  have 

1       rHl(0)      or     C--®I<9).  (i) 

C'  ~ 

It  therefore  follows  that 
(M) 


y  <7r~2' 
^•^ 

m  =  — GO 

This  transcendental  expression,  however,  may  be  expressed  algebraically 

in  terms  of  k. 

i  "HY?/^ 

If  we  write  z  =  -  in  the  formula  z  =  C          > 

we  have 


1    _n     I  W 


_n 

*" 


H[3 


0[3  K  +  ^']          S[K  +  iK']          y(u)l,-oi0 

It  follows  that 

C_lHi(0).  (n) 

*  0i(0) 


THE  FUNCTIONS  sn  u,  en  u,  dn  u.  241 

But  from  (i) 


so  that  C2=i     or     C  =  -L 

*  Vk 

where  the  sign  is  to  be  taken  positive  since  it  is  definitely  determined  from 
the  expression  (M)  above. 
We  thus  have 


If  in  the  integral  of  the  first  kind 

dz 


. -  r 

i/0,l 


-  z2)(l  -  k2z2) 


we  write  z  =  sin 

it  becomes 


Jacobi  *  wrote 

(f>  =  am  u  (amplitude  of  u), 

so  that  .     , 

z  =  sin  <j>  =  sm  am  u. 

If  the  modulus  k  is  zero,  it  is  seen  that  am  u  becomes  u  and  consequently 
z  becomes  sin  u. 

Somewhat  later  z  =  sin  am  u  was  called  the  modular  sine  and  written 

by  Gudermannf 

z  =  sn  u. 

ART.  214.     Consider  next  the  quotient 


We  have  (cf.  Art,  140) 

HI(M)  =  HI(M(Z.S)+  m4K  +  n2iK'] 
Q(u)   "    0[w(z,  s)+m4X  -h  n2i'K/]' 

Since  HI(M)  and  0(i*)  have  the  period  4  K,  it  follows  that 


0(u) 
If  we  take  n  =  1,  we  have 


*  Jacobi,  Werke,  Bd.  I,  p.  81.     Here  Jacobi  retained  the  word  amplitude  of  Legendre 
[Fonct.  Ellip.,  t.  I,  p.  14] 

f  Gudermann,  Theorie  der  Modularfunctionen,  Crelle,  Bd.  18. 


242  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Since  we  have  the  negative  sign  on  the  right,  it  is  well  to  take  the  square 
of  the  quotient,  so  that 


e<td, 

a  formula  which  is  true  for  any  value  of  n. 

ART.  215.  All  the  T  he  ta-f  unctions  have  the  property  of  becoming  zero 
of  the  first  order  upon  only  two  incongruent  points.  It  follows  that  the 
quotient 


0(M) 

becomes  zero  of  the  second  order  upon  two  incongruent  points,  and  upon 
two  incongruent  points  it  becomes  infinite  of  the  second  order. 

Since  HI  fa)  =  0  for  u  =  (2  m  +  l)K  +  n  2  iK', 

it  is  seen  that 

Hifa)  =  0  for  u  =  -  K  and  u  =  -  3  K; 

and  from  above 

0fa)  =  0  for  u  =  -  iK'  and  u  =  -  iK'  -  2  K. 

In  Art.  193  it  was  found  that 

when  u  =  —  K,  then  z  =  —  1, 

when  u  =  —  3  K,  then  z  =  +  1, 

when  u  =  —  iK',  then  z  =00,  s  =  oo, 

when  u  =  —  iK'  —  2  K,  then  z  =  oo  ,  s  =  —  oo  . 


It  follows  from  Art.  150  that        *v  '      is  a  rational  function  of  z.    It  be- 

L©fa)  J 

comes  zero  of  the  second  order  on  the  positions  z  =  —  1  and  z  =  +  1, 
and  infinite  of  the  second  order  on  the  positions  z  =  oo,  s  =  oo  and  z  =  oo, 

S   =  -   oo. 

We  note  that  the  function  z2  —  1  has  the  same  properties.     We  may 
therefore  write 


©fa) 


The  function  Vl  -  z2  is  consequently  like  z  a  one-valued  doubly  periodic 
function  of  u.  It  has  the  period  4  K  but -not  the  period  2  t/sT';  for  when 
u  is  changed  into  u  +  2  ^K',  the  above  quotient  changes  sign.  Hence 
the  other  period  is  4  iK'. 

We  have  

v  1  —  z2  =  vl  —  sn2?/  =  cos  am  u  =  cnu, 

or  cnu-d^M). 

©fa,) 

We  shall  so  choose  the  sign  that  en  u  has  the  value  + 1  when  z  =  0. 


THE  FUNCTIONS  sn  u,  en  u,  dn  u.  243 

This  function  en  u  is  called  the  modular  cosine.     The  analogue  in  trigo 
nometry  is  naturally  the  cosine,  where 


cos  u  =  V 1  —  sin2  u. 

In  order  to  determine  the  constant  Ci,  we  may  write  z  =  0,  S  =  0,  so 
that 

_  c   Hi(0)  c    =   0(0)    =  l-2q  +  2q4-2qg  +  -   •   •      ' 

1  0(0)  Hi(0)  2  t/q  +  2  A/<p  4-  •   •   • 

Again,  if  we  write  z  =  - ,  then,  since  u(-r\  =  —  3  K  -  iK',  it  follows  that 


/7~  ~T  =  c 
V         A'2 


0(-  3  K  -  IK')  0(3  K  +  iK') 

'  0(0) 


But,  since  Ci  =    "  ^  \  ,  we  see  that 

Hi(0) 


#^2  =  y/  -v*  -  "  >,     or    Ci  =  -^-^ 

0(0) 


the  sign  being  definitely  determined  through  C\  = 

In  the  preceding  Article  we  saw  that  Vk  was  definitely  determined  and 
consequently  here  Vk'  is  also  definite. 
We  may  therefore  write 

Vk'  Hiftt) 


ART.  216.    We  saw  in  Art.  152  that  -£•  is  a  one-valued  function  of  u  and 


from   above   it  is    seen  that  Vl  —  z2   is   also   one-  valued.     It  therefore 
follows  from  the  expression 


that  \/l  -  A-2z2  must  be  a  one-valued  function  of  u.     This  function  is 
called  the  cte/ta  amplitude  u  and  written  A  am  u,  dn  u  or  A<£. 

Since  —  =  —  ,  it  follows,  since  z  =  sin  <£,  that  du  =  -&  • 

d*      \/(l-z2)(l-A-2z2)  A0 

To  investigate  this  function  dn  u,  let  us  study  the  quotient 


6(1*)  J     L0(")J 


244  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  zeros  of  the  numerator  are  expressed  through 

u  =(2m  +  l)K  +(2n  +  l)iK'. 
We  may  therefore  take  as  the  two  incongruent  zeros  the  values 

u  =  -  3  K  -  iK'     and     u  =  -  K  -  iK'. 
In  Art.  193  we  saw  that 

u(z,  s)  =  -  3  K  -  iK'  for  z  =  i, 

k 

and  u(z,  s)  =  -  K  -  iK'  for  z  =  -  i. 

k 

Hence  the  above  quotient  becomes  zero  for  z  =  ±  -,  and  it  becomes 

k 
infinite  for  z  =  oo,  s  =  +  QO  and  for  z  =  oo,  s  =  —  <x>. 

The  function  Vl  —  k2z2  has  the  same  zeros  and  the  same  infinities.    We 
may  therefore  write 


Vl  -  k2z2  = 

We  shall  choose  the  sign  so  that  when  z  =  0  the  root  has  the  value  +  1. 
Hence  for  z  =  0  we  have 

r  ®i(0)  r         @(0) 

C2e(oy      C2  =  0^o)' 

If  further  we  write  z  =  1,  we  have 

)       r    @(0) 


It  follows  that  k'  =  C22  or  C2  =Vkf,  and  consequently 

~2q  +  2  q±-  2  q»  +  • 


@i(0)       l+2g  + 

(Jacobi,  Bd.  I,  p.  236.) 


Finally  we  have 


ART.  217.     We  may  write  *  the  three  elliptic  functions  of  u 


(VIII) 


snu  =  — - 

Vk 


*  Cf.  Jacobi,  Werke,  Bd.  I,  pp.  225,  256  and  512;   Hermite,  loc.  cit.,  p.  794. 


THE  FUNCTIONS  sn  u,  en  u,  dn  u. 


245 


The  first  of  these  functions  is  odd,  the  other  two  are  even.     It  follows 
at  once  that 

sn  0  =  0, 

(VIIIO       en  0=1, 
dnO=  1. 

The  zeros  of  sn  u  are     ......     .     2  mK  +  2  niK', 

the   zeros  of  en  u  are     ..    .     ;     .  '     .     (2  m  +  1)K  +  2  niK', 

the   zeros  of  dn  u  are     .    ..     .     .     .     (2  m  -f  1)K  +  (2  n  +  l)t'K'; 

tfi<?   infinities  of  all  three  functions  are     .      2  mK  +  (2  n  +  l)iKf, 

where  m  and  n  are  integers  including  zero. 

We  will  derive  nothing  new  by  forming  other  quotients  of  Theta-func- 
tions. 

ART.  218.     It  follows  at  once  from  the  above  formulas  that 


_ 
Vk  QI(M)      Vk 

ecu) 


dn  u 


or 


/        ,      rr\        C 

sn(u  +  K)= 

dn  u 


We  may  consequently  write 


(IX) 


f         T^N       en  u 

sn(u  +  K)  = 1 

dn  u 

v\           if  sn  u 
cn(u  +  K)  =  —  K ; 

dn  u 

dn(u  +  K}  =  -^-  - 
dn  u 


(IX') 


snK  =  l, 
en  K  =  0, 
dn  K  =  kf. 


When  the  argument  u  is  increased  by  2  K,  it  follows  that 


0(u) 


We  thus  have 


(X) 


sn(w  +  2  K) 
cn(w  +  2  K) 

dn(u  +  2  K} 


sn  w, 
en  u, 

dnu. 


246  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

Noting  that 


/     ,    -j£f\  _  VA/  HI(W  +  iK')  _  V  A/  X(u 
Vk    (d(u  +  iK)        Vk    i\(u 


k  snu 


we  may  write 

(XI) 

and  in  a  similar  manner 
(XII) 
It  is  also  seen  that 

We  thus  have 

(XIII) 


sn(u  +  tTT 


1 


k  sn  u 


k  snu 


sn  u 

sn(u  +  2  iK)  =  sn  u, 
cn(u  +  2  £./£')  =  —  en  u, 
dn(u  +  2iK')  =  -  dnu. 

1 


=   dn  u 

k  sn(u  +  K)       k  snu 

,    .™       dn  u 


cn(u 


Ai.        |        I/A».     y 

A;cn  w 

fr' 

K  +  ^KO  =  -  ^ 


k    cnu 


en  u 


All  three  functions  have  the  periods  4  K  and  4  iK',  so  that 

sn(u  +  4  K)=  snu, 


(XIV) 


and 


cn(u  +  4  K)=  cnu, 
dn(u  +  4  K)=  dnu', 


sn(w  +  4  ijfiL7)  =  sn  u, 

(XV)       cw(u  +  4i7n=cntt, 

c?n  (u  +  4  iKf)  =  dn  u. 

The  periods  of  sn  u  are    .     .     .     .     4  K  and  2  iK', 

the    periods  of  en  u  are     .     .     .     .     4  K  and  2  K  +  2  iK', 

the    periods  of  dn  u  are     ....     2  K  and  4  iK'. 

ART.  219.     The  fundamental  formulas  connecting  the  elliptic  functions 
follow  at  once  from  their  definitions. 

From  the  relations  ^j 

du  =  23j  >          (f>  =  am  u, 


we  have 


du 


THE  FUNCTIONS  sn  u,  en  u,  dn  u. 


247 


It  follows  that 

d  ^^  =  sn'u  =  en  u  dn  u, 
du 

cn'u  =  —  snudnu, 
dn'u  =  —  k2sn  u  en  u. 

The  following  two  relations  are  also  evident  : 

sn2u  +  cn2u  =  1, 
dn2u  +  k2sn2u  =  1. 


Further,  from  the  relations 


—  =  V(l  -  z2)(l  - 
du 


we  have 
and  similarly 


and    z  =  snu, 


-  k2sn2u), 


cn'2u  =  (I  -  cn2u)  (1  -  k2  +  k2cn2u), 
dn'2u  =  (1  -  dn?u)  (dn2u  -  1  +  k2). 
ART.  220.     Jacobi's  imaginary  transformation.*  —  If  we  put 

sin  (j>  =  i  tan  ty, 
it  follows  at  once  that 


sin  <j)  =  i  tan  t/r, 

1 


and  also  that 


If  next  we  write 


cosy1 


sin  T/T  =  —  i  tan  <£, 


COS0 

.    d<f> 
=  -i — ^i 

COS0 


1  -  k2  sin2          v/1  -  k'2  sin2 


r»       ^ 

Jo    Vl  -  k'2s 


=  M*,  say, 
t/u    -v  i  —  K"srn*<p        *so    vi—  /j'^sin->/r 

then  >/r  =  am(w,  A;')  and  $  =  am(iu,  k). 

From  the  relations  above  we  have 


(XVI) 


cn(iu}  k)  = 


'ti   l-'\ 
,u>  *  ; 


u, 


sn(w,  A;')  =  —  i 
cn(u,  k')  = 


cn(iu,  k) 
1 


dn(u,  k)  = 


cn(iu,  k) 

dnd'u.  A:) 
cn(iu,  k) 


*  Jacobi,  Werke,  Bd.  I,  p.  85. 


248 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


ART.  221.     As  a  definition  Jacobi  wrote 

coam  u  =  &m(K  —  u). 


We  have  at  once 


(XVII) 


sin  coam  u  = 


en  u 
dnu 


cos  coam  u  = 


A  coam  u  = 


dn  u 
k' 
dnu 


It  also  follows  that 


(XVIII) 


sin  coam  (iu,  k)  = 


I 


cos 


dn(u,  k') 

ikf 


<MJ  k)  =  —  cos  coam(u,  k'), 
k 

iu,  k)  =  k'  sin  coam(w,  k'). 


ART.  222.     From  the  two  preceding  Articles  it  is  seen  that 
sn  (u  +  iK')  = 


(XIX) 


en  (u  +  iK'}  =  - 


I 

k  sn  u 

idn  u 


-ik' 


k  snu      k  cos  coam  u 
dn(u  +  iK')  =  —  i  cot  am  u  ; 


and  also  that 


(XX) 


sin  coam(w,  +  iK') 


\ 


k  sin  coam  u 

ik' 
k  en  u 

A  coam  (u  +  iK')  =  ik'  tan  am  u. 


cos  coam  (u  +  iK')  = 


ART.  223.     Linear  transformations 
we  put  t  =  kz,  we  have 


/* 


dt 


—  If  with  Jacobi  (loc.  cit.,  p.   125) 


dz 


THE   FUNCTIONS   sn  u,  cnu,  dnu. 
If  further  we  write 

-  -  f         rfz       ., 

Jo  \/(l  -z2)(l  -A--'z2) 


249 


we 


have  z  =  sn(w,  A').    *  =  **(*">  7h  and  consequently* 
if&tt,  -J=  A*  sn(w,  A;), 


(XXI) 


sni 


en 


We  also  have 


dnlku,  -]=  cn(w,  k). 

sin  coam (kit,  -}  = , 

\       kj      sin  coam  (u,  k) 

cos  coam  fjfcu,  -J  =  z'A;'  tan  am(w,  A;), 

A  coam  (ku,  ±]  = ^ 

\       k]      k  cos  am  (u,  k) 


(XXII) 


Next  put  in  in  the  place  of  u  and  observing  that  the  complementary 


modulus  of  -  is  y- ,  it  is  seen  that 

K          K 


(XXIII) 


and 


- 


=  cos  coam  (u,  k'), 


MX- 

ik'\ 
ku,—  \=  sin  coam(w,  A;'), 


(XXIV) 


(ikf\ 
ku,-—\=  cos  am  (u,  kr), 

Iku,  -r"Jsaa  sin  am(?<,  A*'), 


cos  coam 


k 


I        ik'\ 
tan  coamf  &u}  —  -  ) 

V        A;  / 


cot  am  (it,  A/). 

*  See  also  Hermite,  CEuvres,  t.  II,  p.  267. 


250  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

ART.  224.     It  follows  from  Art.  204  that 


and 


Oi(0;  K,iK' 


;  K'  ,  iK) 


Hi(0;  K,iK')        0(0;  K',iK) 

We  have  at  once  (cf.  also  Art.  220) 

.sn(u\K'iK} 

/)/>/•      K         1  K    M    1  X        - 


(XVI) 


1 


cn(u',K',iK) 


cn(u]  K',  iK) 
ART.  225.     Quadratic  transformations.  —  If  we  write 


we  have 


dz 


Mdt 


Z2)(l   -  k 


1 


where 

Writing  u  =    I    —-= 

Jo  v(l 

it  follows  that     (1  +  k)u  =  T 

Jo 
and  consequently 


(XXV) 


In  a  similar  manner  write 


and  M  = 


1 


-f  ksn2(u,k) 


and  we  have 


where 


t  - 
dz 


(1  +  k')z  Vl  -  z2 


Vl  -  k2z2 


Mdt 


-  z2)(  1  -  A;2z2) 

1    _  j,r 


1  +  k' 


and   M  = 


-  t2)(  1  - 


k' 


THE   FUNCTIONS   sn  u,  cnu,  dnu. 

It  follows  at  once  that 


251 


(XXVI) 


en 


(1  4-  k')sn(u,  k)cn(u.  k} 
dn(u,  k) 

I  -(1  +kf)sn2(u,k) 

dn(u,  k) 
1  -  (1  -  k')sn2(u,  k) 

dn(u,k) 


In  formulas  (XXVI)  change  A;  to  l/k  and  w  to  uk  and  observe  formulas 
(XXI).     It  is  seen  that 


(XXVII) 


sn(k 


+  tfc'J  cn(u,  k) 

-  ik'1  =  I  -(k  +  ik')ksn2(u,k) 
+  ikr \  cn(u,  k) 

-  &1  _  1  ~  (k  -  ikf}k  sn2(u,  k) 

cn(u,  k) 


The  formulas  just  written  are  the  very  celebrated  formulas  due  to 
John  Landen  (Phil.  Trans.,  LXV,  p.  283,  1775;  or  Mathematical  Memoirs, 
1,  p.  32,  London,  1780)  and  may  be  derived  as  follows: 

Write 

*  sin  (2<f>  —  <j>i)=  ki  sin  0i,  (1) 


where 


,         1-k' 

*1  =  T— 77' 


i  <  k, 


Since 

it  is  evident  that 

.sin  (20  —  0i)<  sin 

(20-0!)<0!, 
0<   01- 

Solving  (1)  for  0,  we  have 


sin220  =(1  +£1)2sin201 


[~l  - 


sin*  0 


]- 


or,  since  — 

it  is  seen  that 

We  further  have 


A0 


(1  4-  A-' 


!       \/!-Jfc2sin20 


252  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

ART.  226.  Development  in  powers  of  u.  —  If  we  develop  by  Maclaurin's 
Theorem  the  three  functions  sn  u,  en  u,  dn  u,  we  obtain  the  following 
series: 

snu  =  u-(l  +  &2)^ 
o  ! 


dnu  = 


U2n+l  y2n 

where  the  coefficient  of  any  term,  say  -  -  —  or  -  -  —  ,  is  an  integral 

(2n  +  1)!       (2n)l 

function  of  k2  with  integral  coefficients. 

Following  Hermite*  we  wish  to  determine  these  coefficients.  From 
the  formulas  derived  above 

sn  (  ku.  -}=  k  sn(u.  k), 
\       k/ 

en  (ku,  -  }  =  dn(u,  k), 
\       k/ 

it  is  seen  that  the  coefficients  of  sn(u,  k)  are  reciprocal  polynomials  in  k 
and  that  those  of  dn(u,  k)  may  be  derived  immediately  from  those  of 
cn(u,  k). 

Gudermann  •)•  has  shown  that  the  coefficients  of  en  u  are' 

1  +  4  k2, 

1  +  44  k2  +  16  A;4, 

1  +  408  k2  +  912  fc*  +  64  k6, 

1  +  3688  k2  +  30768  A;4  +  15808  k6  +  256  k8, 

We  note  that  if  we  put  k  =  cos  0  and  introduce  the  multiple  arcs  instead 
of  the  powers  of  the  cosines,  the  above  coefficients  when  multiplied  by 
k  may  be  written 

k  +  4  A;3  =  4  cos  6  +  cos  3  0, 

k  4.  44  yfc3  +  16  k5  =  44  cos  0  +  16  cos  3  0  +  cos  5  6, 
k  +  408  &3  +  912  k5  +  64  k7  =  912  cos  0  +  408  cos  3  0  +  64  cos  5  6  +  cos  7  6, 

In  these  equalities  it  is  seen  that  the  powers  of  k  and  the  cosines  of  the 
multiples  of  6  have  precisely  the  same  coefficients. 

*  Cf.  Hermite,  Comptes  rendus,  t,  LVII,  1863  (II),  p.  613;  or  (Euvres,  t.  II,  p.  264. 
t  Gudermann,  Crette,  Bd.  XIX,  p.  80. 


THE   FUNCTIONS   sn  u,  en  u,  dn  u.  263 

*  u2n  +  2 

In  general,  if  we  denote  the  coefficient  of  —  -  — 

\2i  TL  ~r~  —  )  I 

by 

t  =  n 

A0  +  A,  k2  +  A2k*  +  -   •  -  +  Ank2n  -  ]£  Aik2*  =  cn<2"+2>  (0,  fc), 

t  =  0 

we  will  have  the  relation 


which  may  be  demonstrated  as  follows: 
From  formulas  (XXVI)  we  have 

|~,7    ,    ..,.      k  -  ik*~\      I  -  (k  +  jkf)ksn2(u,k) 
cn\  (k  +  ik')u,  -  -  —    =  -  *  -  —*—  —  ^-^, 
L  k  +  ik']  cn(u,k) 

and  changing  i  to  —  i  it  follows  that 

cn\(k  _  *>,  *±*H  =  i-(*-v*o*«»'(u.fe). 

k  —  IK  J  cn(u,  A;) 

From  these  two  formulas  it  follows  at  once  that 

(k  +  ik'}cn\(k  -  ik')u,  L±jK\  +  (k  -  ik')cn\  (k  +  ik')u,  k  ~  ik/ 
|_  fc  —  ik'_\  |_  k  +  ik' 

=  2kcn(u,k). 
In  this  formula  write  k  =  cos  0,  fc'  =  sin  0,  and  we  have 

e*cn(e-"w,  e2i0)  4-  e-i9cn(0*u,  e~'2le]  =  2  cos  0  cn(w,  fc). 
Noting  that 


it  is  seen  by  equating  the  coefficients  of  -  —  on  either  side  of  this 
equation,  when  expanded  by  Maclaurin's  Theorem,  that 
2AiC082t+lO  =  S^-cos  (2n  +  1  --  4i)0. 

From  this  formula  the  quantities  J.0  =  1.  AI,  A-?,  .  •  .   ,  may  be  determined 
at  once. 

For  example,  let  n  =  4  and  for  brevity  put  At  =  4l'az.     If  the  multiple 
arcs  are  replaced  by  the  powers  of  the  cosine,  we  have 

cos  0  +  4  a  i  cos30  +  16  a2  cos50  +  64  a3  cos"0  +  256  a4  cos90 
=  cos  0  +  ai(cos  3  0  +  3  cos  0)  +  a2(cos  5  0  +  5  cos  3  0  +  10  cos  0) 
+  a3(cos  7  0  +  7  cos  5  0  4-  21  cos  3  0  +  35  cos  0) 
+  a4(cos  9  0  +  9  cos  7  0  +  36  cos  5  0  4-  84  cos  3  0  -4-  126  cos  0) 
=  cos  9  0  +  4  a!  cos  5  0  4-  16  a2  cos  0  +  64  a3  cos  3  0  +  256  a4  cos  7  0. 


254  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

We  thus  have  among  the  a's  the  five  equations  * 

1  -  a4, 

4«i  =  a2  +  7a3  +  36  a4, 
16  a2  -  1  +  3  ai  +  10  a2  +  35  a3  +  126  a4, 
64  a3  =  a!  +  5  a2  +  21  a3  +  84  a4, 
256  #4  =  #3  +  9  a4. 

Since  the  sum  of  these  equations  leads  to  an  identity,  we  may  omit  any 
one  of  them,  say  the  third;  and  from  the  other  four  we  have 

«i=  922,         a2=  1923,         o3=  247,         a4  =  1, 

which  agree  with  the  above  results  of  Gudermann. 
Since 


the  coefficients  of  dn(u,  k)  are  at  once  deduced  from  those  of  cn(u,  k); 
while  those  of  sn(u,  k)  may  be  obtained  from  the  formula 

sn'(u,  k)  =  cn(u,  k)  dn(u,  k). 

[See  Table  of  Formulas,  LVIL] 

DEVELOPMENT  OF  THE  ELLIPTIC  FUNCTIONS  IN  SIMPLE  SERIES  OF 
SINES  AND  COSINES.    • 

First  Method. 
ART.  227.     In  Art.  206  we  saw  that 


Noting  that 


A=l 

^  t* 

log(l  +0-  --2,(-V  AT> 

A=l  X 

and  that 

1  -  2qcos2u  +  g2  =  (1  -  qe2iu)  (1  -qe~2iu), 

it  is  seen  that 

-  1  ,  9.  92  cos  4  w 

—  -  log  (1  —  2  g  cos  2u  +  q2)  =  q  cos  2  w  +  -  —  ^~- 

2  2i 


(f  cos  6  u      q4  cos  8  u 
~~~  ~~~ 


THE  FUNCTIONS  sn  u,  cnu,  dnu.  255 

We  therefore  have 

=  const.  -  cos2u(q    +  q3    +  <f    +  •••) 


or  1  .      -    2  #iA  9  cos  2  M.      g2  cos  4  u 

const.  -  <L__  -  ___ 

3  cos  6  w        4  cos  8  u 


1  .      -  /2  #iA 
-  Iog0(—  ) 


3(1  -  ?6)       4(1  -  g8) 

The  logarithms  of  the  other  Theta-f  unctions  may  be  expressed  in  a  similar 
manner. 

ART.  228.     Hermite  (CEuvres,  t.  II,  p.  216)  gives  the  following  method 
for  the  expressions  of  sn  u,  en  u,  dn  u  in  terms  of  the  sines  and  the  cosines. 

We  have  the  formulas 

,  _  d  log  (dn  u  —  k  en  u} 

K  STl  U  —  •  y 

du 

.,  _  d  log  (dn  u  +  ik  sn  u) 

IK  CTl  U  —  •  i 

du 

idnu  =  d  log  (cn  u  +  ?'  sn  u^  • 
du 

We  shall  next  derive  the  formulas 


=  1-2  Vq  cos  u  +  q     1-2  \/g3  cos  u  +  q3     1-2  Vq5  cos  u  +  g5 
1+2  Vg  cos  ?^  +  5     1+2  Vq3  cos  u  +  q3     1+2  X/g5  cos  w  +  q5 


_  1-2  V-qsin  u-q     1-2V  -q3  sinu  —  q3     l—2V  — 
l+2\/-qsmu  —  q     1  +  2V  —  q3  sinu  —  q3 

,0.  2  Ku   ,    .       2  Kt* 

(o)         en  --  h  t  s^i  - 


Jacobi  [Werke,  I,  p.  143,  formula  (5)]  has  implicitly  derived  formulas  (1) 
and  (2)  above,  the  first  being  had  when  in  Jacobi  's  formula  u  is  changed 

to  ^  —  u,  and  the  second  when  —  q  is  written  for  q. 


256  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

These  two  formulas  may  be  derived  directly  in  the  manner  which  we 
now  give  for  the  formula  (3)  above,  ^u 

Write  as  in  Art.  205,  $(u)=  1  -  e* ";  the  expression  which  we  wish  to 
demonstrate  equal  to 

en  u  +  i  sn  u 
will  take  the  form 

+  3  iKf)  <b  ( —  u  +  5  iK') . 


Multiplying  numerator  and  denominator  of  this  expression  by 

A<j>(-  u  +  iK')  </>(u  +  3  iK')</)(-  u  +  5  iK') 
where  A  is  a  constant,  and  putting 

xiu 

<b(u)  =  Ae2K</>2(~  u  +  iKr)^(u  +  3iK')cj>2(-  u  +  5iK') 
we  have  to  demonstrate  the  formula 

en  u  +  i  sn  u 


©  (u) 
We  further  note  that 

$>(u  +  2/0  =  - 
$u  +  4  i/T  -  $W 


The  same  functional  equations  are  satisfied  by  H(w)  and  HI(W).  In 
Art.  90  it  was  shown  that  any  three  intermediary  functions  of  the  second 
order  were  connected  by  a  linear  relation,  so  that  here  we  may  write 


Divide  this  expression  byO(w),  and  we  have 


-  u  +  3  iK')  (f>(u 

=  D  en  u  +  iB  sn  u. 

Writing  u  =  0  and  u  =  K  respectively  in  this  formula  we  have  D  =  1  and 

B  =  1,  which  we  wished  to  demonstrate. 

From  the  formulas   (1),   (2)  and   (3)  we  have  (see  Jacobi,  Werke,  II, 

p.  296) 

kK      2Ku  _  Vq  sin  u      Vq3  sin  3  u      Vf  sin  5  u 
2*S>       TT  1  -q  1  -<?  1  -36 

kK      2  Ku  _  \/qco$u      Vq3  cos  3  u      Vg5  cos  5  u 
'    — 


—  ri    ^  ^U  —  I  _i_  g  CQS  2  u      q2  cos  4  w       q3  cos  6  w 
2ffai      „      -  4  +   1+52          j  +g4          i+?6 


THE   FUNCTIONS   sn  u,  en  w,  dn  u.  257 

Second  Method. 

ART.  229.  Suppose  with  Briot  and  Bouquet  (Fonct.  Ellipt. ,  p.  286)  that 
f(u)  is  a  doubly  periodic  function  of  the  2  nth  order  with  periods  4  K  and 
2iK'  such  that  f(u  +  2K)=-f(u) 

and  further  suppose  that/(w)  has  n  infinities  ah  within  (see  Art.  91)  the 

period-parallelogram  ABDC,  where  A  is  an  arbitrary  point  u0,  while  B  and 

C    are    the    two    points    u0+2K    and   w0+2tK'. 

Form  the  parallelogram  EFGH  whose  vertices  E  and 

#are  the  points  M.O  -  2  ?rcW  and  u0  +  2  m'iK',  while 

F  and  (7  are   the   points   u0  +  2  K  -  2  m'iK'  and 

u0  +  2K  +  2  m'iK'.     The  infinities  of  /(M)  situated 

within  the  parallelogram  EFGH  may  be  represented 

by  a  =  ah  +  2  miK' ,  where  w  varies  from  —  m'  to 

ifi'-J. 

Let  t  be  any  point  situated  within  this  parallelo 
gram.     The  function 

/(«) 


has  the  period  2  #;  its  poles  are  the  point  *  and  the  points  a  =  ah  +  2  miK'. 
It  follows  from  Cauchy's  Theorem  that  the  definite  integral 

—  du, 

t) 


j_  r     /(«*) 

••y  **«••- 


where  the  integration  is  taken  over  the  sides  of  the  parallelogram  EFGH, 
is  equal  to  the  sum  of  the  residues  relative  to  the  poles  that  are  situated 
within  this  parallelogram.  The  two  sides  FG  and  HE  give  values  that 
are  equal  and  of  opposite  sign,  while  on  the  sides  EF  and  GH  the  function 
f(u)  has  a  finite  value  and  mod.  -  -  tends  towards  zero  *  when 

m'  becomes  very  large.  sm  ^K  ~  ^ 

Thus  when  m'  becomes  very  large  the  definite  integral  tends  towards 
zero  and  consequently  the  sum  of  the  residues  is  zero. 

The  residue  relative  to  t  being  —/(*),  we  have  the  equation 


*  In  write  u  =  x  +  iy  and  note  that 

sm  u 

1 

2? 

sin  M 

e^Tw  _  e-ix±v 

=  0  f  or  v  =  oo . 


268  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

If  f(u)  has  only  simple  infinities,  which  case  alone  is  necessary  for  our 
investigation,  the  above  equation  becomes 

Ah 


sin  —  (t-ah-2  miK') 
2  K 

where  A  h  is  the  residue  of  f(u)  relative  to  an*  The  series  is  convergent 
in  both  directions.  This  equality  is  thus  demonstrated  for  all  points  t 
situated  within  two  indefinitely  long  parallel  lines  EH  and  FG.  Since 
both  sides  of  this  equation  change  signs  when  t  is  replaced  by  t  +  2  K, 
the  equality  is  true  for  all  values  of  t',  and  consequently  we  have  for  the 
finite  portion  of  the  w-plane 


=  -00/1=1  sin——  (u  —  ah  —  2  miK') 
2  K 


ART.  230.  Consider  next  a  doubly  periodic  function  f(u)  with  periods 
2  K  and  2  iK'  and  having  n  infinities  ah  within  the  parallelogram  ABDC 
of  the  preceding  Article. 

The  function  f(u) 


admits  the  period  2  K,  and  the  definite  integral 

du 


relative  to  the  contour  of  the  parallelogram  EFGH  is  equal  to  the  sum  of 
the  residues  with  respect  to  the  poles  situated  within  the  parallelogram, 
that  is,  for  the  point  t  and  the  points  a  =  ah  +  2  miK',  where  m  varies  from 
—  m'  to  m'  —  1.  The  sides  FG  and  HE  give  equal  results  with  contrary 
sign.  If  we  represent  by  u  a  point  on  the  line  A  B,  the  congruent  points 
on  HG  and  EF  are  u  +  2  m'iK'  and  u  —  2  miK',  and  the  parts  of  the 
integral  relative  to  these  two  sides  are 

1 1 

(u-t-2  m'iK')       tan^  (u  -  t  +  2 
2  K 

When  m'  becomes  very  large  the  first  tangent  tends  towards  --  i  (see 
Art.  25)  and  the  second  tangent  towards  i,  so  that  the  integral  just  written 
tends  towards  a  limit  equal  to  the  rectilinear  integral 


1      (*un  +  2K 

M=~   I          f(u)du 

K  Ju0 


THE  FUNCTIONS   sn  u,  en  u,  dn  u.  259 

along  the  line  AB.     The  residue  of  the  function  relative  to  the  point  t 

9  TV- 
being  f(t),  we  have,  as  in  the  preceding  Article, 


and  consequently  if  the  function  has  only  simple  infinities 

m=  +  M    h  =  n 

Ah 


m=  -x   h  =  i  tan  — —  ( 

where  t  is  any  point  in  the  finite  portion  of  the  w-plane,  and  Ah  is  the 
residue  of  f(u)  relative  to  ah. 

ART.  231.  To  make  application  of  the  results  of  the  two  preceding 
Articles  consider  the  ratios  of  the  four  Theta-f unctions.  Of  these  twelve 
ratios  eight  satisfy  the  relation  f(u  +  2  K)  =  —  f(u)  and  four  the  relation 


f(u  +  2K)  =  f(u).     Take  the  two  functions  ^v  and  ^-^ .    Form  a  paral- 

H(M)  H(M) 

lelogram  EFGH  with  the  origin  as  center  and  vertices  ±  K  ±  (2  m'  +  l)iK'. 
The  infinities  of  these  two  functions  are  the  zeros  of  H(w).  Those  infinities 
within  the  parallelogram  are  represented  by  the  formula  a  =  2  miK', 
m  varying  from  —  m?  to  +  m' ;  all  these  infinities  are  simple. 

The  residue  of  "  ^u'  relative  to  the  .infinity  2  miK'  is     ,    '  ;  that  of     x^u^ 


s  _ 


H'(0) 
We  therefore  have 


m 

v  ; 


*  ei(0)mi 


H(u)       2K  H/(0)m=_oogm_._ 

2K 

Replacing  hi  these  two  formulas  u  by  the  quantities  u  +  K,  u  4- 
+  K  +  iKf  we  have  six  additional  formulas  including 


*   ©(O)"1 


2K 

m=  4-oc 


-  D 


_Msi          [u  _  (2m  _  x 
2  A. 


260  THEORY   OF   ELLIPTIC    FUNCTIONS. 

To  develop  the  function  1*      ,  say,  which  admits  the  period  2K,  we 

apply  the  method  of  the  preceding  Article.  We  note  that  for  congruent 
points  on  the  sides  EF  and  GH  of  the  parallelogram  EFGH,  the  difference 
of  the  values  u  being  equal  to  (2  ra'  +  1)2  iK',  the  function  f(u)  takes 
equal  values  with  contrary  signs;  and  the  values  of  the  tangent  on  these 
two  sides  being  ^  i,  the  definite  integral  relative  to  these  two  sides  is 
zero. 

We  therefore  have 

et(u)     «•  H^o 

0(«)        2K  H'(0) 

2  K 

Further,  since 


we  have  by  differentiating  sn  u  with  regard  to  w,  and  then  writing  u  =  0 


«  -;  and  since      l       «  %       , 
H'(0)       V/c  0(0)        v  A;' 


5lffli  =  JL  and  similarly  M  -  -4=- 
Hr(0)       x//cr  Hr(0)       V^A;' 


It  follows  immediately  from  (3),  (4)  and  (5)  that 

m—  +00 
IR\  ™  v  -       ^        ^ 

(0;  sn  u  —  —— 


__OCgin_[M  _(2m  -  l)iK'] 
2  7£ 


"" 


2  K 
(8)  *""- 


If  we  group  the  terms  two  and  two  the  equations  (6)  and  (7)  become 

27iVo  .     nu 


(9) 


27: 

do) 


2K  +q 


*xu    .    ^m-2 

2K      a 


THE   FUNCTIONS   sn  u,  en  u,  dn  u.  261 

The  series  (8)  is  not  convergent  in  both  directions;  but  if  from  dn  0  we 
subtract  dn  u,  we  have  the  convergent  series 

1    J_  n2m-l 


(11) 


_  r, 


f\     TS- 

Observing  that 


q  Sl°  3  '  +  I"  sin  5  ' 


it  is  evident  that  (9)  and  (10)  may  be  written 


m=  1 


These  values  are  the  same  as  those  given  at  the  end  of  Art.  229,  where 
the  corresponding  value  of  dnu  is  found. 

By  considering  the  quotient  !L        as  given  in  equation  (1)  and  also  the 

H(uj 

quotient  —  ^  ,  we  may  derive  in  a  similar  manner 
HI(U) 


m  =  l 

[See  Jacobi,  Werke,  I,  p.  157.] 
EXAMPLES 


1.  Prove  that  *n(iu  +  K}  =  ~^-  • 

2.  Show  that 

ikf  sin  am  u 


in  amf  z'A/w.  —  ) 
\         Ar  / 


sm        . 

cos  am  u 

I         1  \       A  am  u 

cos  ami  ik'u,  —  }  = , 

K  I      cos  am  u 

A  amhfc'w,  — )  = 

\         k'/      cos  am  u 


262  THEORY   OF   ELLIPTIC   FUNCTIONS. 

3.   Show  that 


am  u 


4.   Prove  that 


(ik'\      tk  sin  am 
iku,  — )  =  — 
k  I         A  am  u 

I  i      ik'\           ! 
cos  am  [  iku,  —  ]  = , 

\         k  I     A  am  u 

I .,      ik'\      cos  am  u 

A  am  ( iku,  —  ]  =  — 

\         k  J       A  am  u 


1 


sn2(iu,  k)      sn2(u,  k) 


5.   Derive  the  formulas 


' 


cn(l 


dn(u,k) 


Suggestion  :  apply  formulas  (XVI)  to  formulas  (XXV). 
6.  Show  that 

~  (kf  +  ife)  gn(M,  fe)  dn(u,  fc) 


cn\  (kf  +ik)u, 
dn\  (kf  +  ik)u, 


1  —  (k  —  ik')k  sn2(u,  k) 
2  Vikk'  1  cn(u,  k) 

i /j,   i  v^/^  i*  o^>2^/  i-\ 

X  yA/        I^     c/A/    ^   A/   o/t'     ytC^    A/y 

A;'  +  ik  J  =  1  -  (k  -  ik')  k  sn2(u,  k) ' 


7.   Show  that 


sn\  (k  -ik')u, 


cn\(k-ik')u, 


dn\(k-  ik')u, 


k  +  ik' 
k  -  ik' 
k  +  ik'' 

k  +  ik'' 
k  -  ikf 


8.   Show  that 


9.   Show  that 


H'(it)  =fc  en 


(k  —  ik'}  sn(u,  k)dn(u,  k) 

cn(u,  k) 
l-(k-  ik')k  sn2(u,  k) 

cn(u,  k) 
1  -(k  +  ik')ksn2(u,k) 

cn(u,  k) 


u}  +Vk  snu®'(u). 


0(0) 


0(10  ' 


THE  FUNCTIONS  sn  u,  en  u,  dn  u. 

10.    Prove  the  following  relations : 

0(0,' k)       VP6'  0(0,  fc')       ' 

H(m,  k)        .  Ik  _*&  H(w,  fc') 


_ 
0(0,  A;)  fc'  0(0,fc')' 

K,fr)  _    /fc"    ^©(M  +K', 
,  Jk)  VA;/e  0(0,  A:') 


0(0 


11.   Show  that 
and  that 


— 

du        snu 


w  +  iK'}-dn2u 


d  /snu  dnu\          2  2 

—  (  -   =  dn2u  +  dn2(iu.  fc')—  1- 
rfw  \     en  M     / 


12.    Prove  that  sn  u  dn"u  —  sn"u  dnu  =  snudnu; 
and  that 

(sn  w)2,  sn  u  sn'u,  (sn'w)2 


(en  u)2,  en  u  cn'u,  (cn'u)2 
(dn  u}2,  dn  u  dn'u,  (dn'u)' 


kf  snucnu  dn  u. 


263 


13.   Show  that 
2kK 


(G.  B.  Mathews.) 


cos  coam 

R 

14.   Show  that 

2  k  K       1 
-          2X7*  r 

r 


2  A"w      4  Vg sin  w      4 \/q3  sin  3  M      4  \  cf  sin  5  w 
«  1  +q  1  +53 


4  q  4<f  4o3 

1  — ^  cos  2  u  + :  cos  4  w  — z  cos  6  u  +  •  •  •  . 


CHAPTER   XII 
DOUBLY  PERIODIC  FUNCTIONS  OF  THE  SECOND  SORT 

ARTICLE  232.  From  the  formulas  (X)  and  (XII)  of  the  preceding 
Chapter  it  follows  that  dn  u  has  the  period  2  K  and  sn  u  the  period  2  iK', 
although  2  K  is  not  a  period  of  sn  u  and  2  iK'  is  not  a  period  of  dn  u. 
There  is  consequently  an  irregularity  in  this  respect.  In  order  fully 
to  understand  this,  it  is  well  to  consider  the  doubly  periodic  functions 
of  the  second  sort  which  were  introduced  by  Her  mite.* 

The  Germans  use  the  word  "Art"  for  the  word  "espece"  which 
I  translate  by  "  sort  "  (see  Art.  84  where  the  doubly  periodic  functions 
of  the  third  sort  were  treated  under  the  name  "Hermite's  intermediary 
functions  ").  In  this  connection  see  Jordan,  Cours  d' Analyse,  t.  II, 
No.  401,  and  Halphen,  Traite  des  fonctions  elliptiques,  t.  I,  pp.  325-338, 
411-426,  438-442,  463. 

ART.  233.  A  doubly  periodic  function  of  the  second  sort  with  the 
primitive  periods  2  K  and  2  iK'  is  denned  through  the  functional  equa 
tions 

f(u  +  2K)=  vf(u), 

f(u  +  2iK')=  v'/(iO, 

where  v  and  i/  are  constants  called  factors  or  multipliers  and  are  inde 
pendent  of  u.  When  v  =  1  =  i/,  we  have  the  doubly  periodic  functions 
properly  so  called,  which  belong  to  the  category  of  doubly  periodic 
functions  of  the  first  sort. 

In  the  case  before  us  of  the  preceding  Article  sn  u,  en  u,  dn  u  belong 
to  the  class  of  functions  of  the  second  sort,  as  appears  from  the  formulas 
(X)  and  (XII). 

For  the  function  sn  u  we  have  v  =  —  1,  i/=  1;  for  en  u  we  have 
v  =  _  1;  i/=  —  1,  while  v  =  +  1,  i/=—  1  for  dn  u.  We  may  now 
consider  more  closely  these  doubly  periodic  functions  of  the  second 
sort. 

*  Hermite,  Comptes  Rendus,  t.  53,  pp.  214-228,  and  t.  55,  pp.  11-18  and  pp.  85-91; 
Hermite,  Note  sur  la  theorie  des  fonctions  elliptiques,  in  Lacroix's  Calcul,  t.  2  (6th  ed.), 
pp.  484-491;  see  also  Cours  de  M.  Hermite  redige  en  1882,  par  M.  Andoyer,  p.  206; 
Appell,  Ada  Math,,  Bd.  13,  1890;  Picard,  Comptes  Rendus,  t.  90,  pp.  128-131  and 
293-295;  Picard,  Crette,  Bd.  90,  pp.  281-302;  and  in  particular  Forsyth,  Theory  of 
Functions,  pp.  273-281,  where  references  are  made  among  others  to  Frobenius,  Crelle, 
Bd.  93,  pp.  53-68;  Brioschi,  Comptes  Rendus,  t.  92,  pp.  323-328. 

264 


DOUBLY   PERIODIC    FUNCTIONS    (SECOND    SORT).         265 

ART.  234.  Formation  of  the  doubly  periodic  functions  of  the  second 
sort  which  have  prescribed  factors  v  and  i/.  —  In  the  following  Article  it  is 
shown  that  it  is  always  possible  to  form  a  fundamental  doubly  periodic 
function  of  the  second  sort  f(u)  with  factors  v  and  i/,  which  function  is 
infinite  of  the  first  order  at  only  one  point  within  the  parallelogram 
with  sides  2  K  and  2  iK'  '.  The  infinity  of  this  fundamental  function 
is  denoted  by  u  =  c. 

This  admitted  for  the  moment,  let  F(u)  be  an  arbitrary  doubly 
periodic  function  of  the  second  sort  which  has  the  periods  2  K  and.  2  iK' 
and  has  the  same  factors  v  and  i/  as/(w).  Further  we  shall  assume  that 
F(u)  is  determinate  at  every  point  of  the  period-parallelogram. 

Suppose  that  the  function  F(u)  is  infinite  of  the  k  order  at  the  points 
a.i  (i  =  1,  2,  .  .  .  ,  ri)j  where  the  points  0.1,  a2,  .  .  .  ,  an  all  lie  within  the 
period-parallelogram. 

We  shall  show  that  F(u)  may  be  expressed  in  terms  otf(u). 

For  simplicity  suppose  that  the  parallelogram  is  so  situated  (Art.  91) 
that  F(u)  does  not  become  infinite  upon  its  sides. 

Consider  next  the  function 


where  u  is  any  point  within  the  period-parallelogram,  while  £  is  to  be 
regarded  as  the  independent  variable. 

Instead  of  c  write  £  +  2  K.     It  follows  that 

t(,-  +  2  K)  =  F(£  +  2  K)f(u  -  c  -  2  K). 
But  we  have  f(u  +  2  K)=  vf(u), 


If  we  put  £  +  2  K  for  £  in  this  last  formula,  the  result  is 


v 

Also,  since  F(£  +  2  K)  =  vF(£), 

it  follows  that 

^(£  +  2/0=F(£)/ 

or  +(*  +  2K)=  VT(£), 

and  similarly 


It  is  thus  seen  that  ^(£)  is  a  doubly  periodic  function  of  the  first  sort. 
For  such  a  function  we  have  proved  that 


where  the  summation  is  to  be  taken  over  all  the  infinities  within  the 
period-parallelogram. 


266  THEORY   OF    ELLIPTIC    FUNCTIONS. 


But  ^r(£)  becomes  infinite  on  the  points  where  F(£)  is  infinite  and 
besides  on  the  point  u  -  £  =  c,  where  f(u  —  £)  is  infinite.  The  points 
a  i,  a2,  .  .  .  ,  oin  must  be  distinct  from  the  point  u  —  c  =  £. 

The  expansion  of  f(u  —  £)  in  the  neighborhood  of  the  point  c  is  of 
the  form 


f(u- 


-  (u  -  c) 


(In  the  sequel  we  shall  choose  a  fundamental  function  f(u)  such  that 
the  quantity  C  is  unity.) 

Next  if  we  develop  F(£)  in  the  neighborhood  of  u  —  c  by  Taylor's 
Theorem,  we  have 

F(0  -  F(u  -  c)  +  F'(u  -C)[t-(u- 
and  since 


we  have  Res  ^r(£)  =  -  CF(^  -  c). 


In  the  neighborhood  of  the  infinity  ak,  the  expansion  of  F(£)  is  of  the 
form  (cf.  Art.  98) 


while  the  expansion  of  f(u  —  £)  in  the  neighborhood  of  this  point  is 
f(u  -f)-/(u  -a.)-/-^T^(f-  a^  +  ^'^-^ff 


Through  the  multiplication  of  these  series  it  is  seen  that 

Res  TK£)  -  At,  i/(«  -  «»)  -  ^  /'(«  -  «*)  +  4lr  / 
e-ajk  2! 


Since 
we  have 


DOUBLY   PERIODIC   FUNCTIONS    (SECOND    SOKT).         267 
If  next  we  write  u  +  c  in  the  place  of  u,  it  follows  that 

k  =  n 

CF(u)  =         Ak,  i/(u  +  c  -  a,)  -  2*&f'(u  +  c  -  ak) 


+  •   •   •  ±  ,'        /(A*-1}("  +  c  -  a 
(*k  -  1)! 

which  is  the  expression  of  F(u)  in  terms  of  the  fundamental  function 

/(«). 

ART.  235.  Formation  of  the  fundamental  function  f(u)  which  has 
prescribed  factors  (or  multipliers)  v  and  v'  ',  where  v  and  vf  are  any  constants 
different  from  zero. 

We  had  the  formulas 


2iK')  =  - 

-  ^(u+iK") 

where  JJL  =  ft(u)  =  e 

If  we  write 

0(u)=H(u  +  /?), 
it  follows  that 

<j>(u  +  2K)  =  H(u  +  ^  +  2K)  =  -  H(u 
or, 

<f>(u  +  2K)=-  <f)(u)t 
and  similarly 

_7nl 

4>(u  +  2iK')  =  -  ae     K  >(M). 
Consider  next  the  function 


H(M)          H(M) 
We  have  immediately 


- 
V(u  +  2iK')=  V(u)e     K  . 

The  function  ^(u)  is  therefore  a  doubly  periodic  function  of  the  second 

_^ 

sort  having  as  factors  +1  and  e     K  .     Suppose  that  v  and  i/  are  the 
prescribed  factors.     To  form  a  function  having  them,  write 

A«)-«flM*(«); 

so  that 

f(u  +  2K)=  ea(»+2*)  ¥(M  +  2  K)  =  ea**f(u) 
and 

_r^t 

/(w  +  2iK')=ea(u+2ijm&(u  +  2iK')=ea2iKfe     *f(u). 
Hence  f(u)  is  a  doubly  periodic  function  of  the  second  sort  with  the 

a  2  j  jf  _  ^ 

factors  ea2K  and  e  K  . 


268  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  arbitrary  constants  a  and  /?  may  be  so  chosen  that 
(1) 

(2)  e 

From  (1)  it  follows  that 

and  from  (2) 


or 

=  g'logy  +  g^logi/^ 

7T 

The  quantities  a  and  /?  being  thus  determined  we  have 


The  function  f(u)  is  infinite  of  the  first  order  for  u  =  0  (see  Art.  203) 
and  for  no  other  point  in  the  period-parallelogram,  since  the  other 
vertices  of  the  parallelograms  are  counted  as  belonging  to  the  following 
parallelograms. 

ART.  236.  There  is  one  case  *  in  which  we  cannot  determine  f(u)  in 
the  above  manner,  viz.,  when  the  multipliers  or  factors  v  and  i/  have 
been  so  chosen  that 

/?  =  2  mK  +  2  niK', 

where  m  and  n  are  integers. 
We  would  then  have 

f(u)  =  eau  K(u 


HO) 

(M  +  2niK') 


Further,  since  (cf.  Art.  91) 

H(u  +  n2iK')=  (  -  l)n  e~  *' 

it  follows  that 

«H(1 


so  that  f(u)  is  an  exponential  function  and  no  longer  a  doubly  periodic 
function  of  the  second  sort. 

*  See  Forsyth,  Theory  of  Functions,  p.  279. 


DOUBLY   PERIODIC    FUNCTIONS    (SECOND    SORT).         269 

ART.  237.     We  must  proceed   differently  for   this  exceptional  case. 
We  had  by  hypothesis 

/?  =  2  mK  +  2  niK', 
and  consequently 

2  mKn  +  2  nK'in  =  K'  log  v  +  Ki  log  i/. 
Further,  since  log  v  =  2  Ka,  it  follows  that 

#i  log  *'  =  2  mK-  +  2  nK'ix  -  2  KK'a, 

or  log  i/  =  -  2  ra-i  +  2n  —  ~  +  2  K'cn. 

/l 

We  thus  have 

v'  = 
and 

v  =  e2jfiro:  =  e 
If  we  put 

mrt 

«  -  ~K 
the  above  expressions  become 


We  have  the  exceptional  case*  when  v  and  i/  have  this  form.  The 
quantity  7-  is  arbitrary;  but  if  the  factors  v  and  i/  are  given,  then  7-  is 
known. 

We  now  write 


where  H'^)  is  the  derivative  of  H(u). 
From  the  formulas 


HCv)-  H'(u)J- 


2K)=-  H(w), 
we  have  at  once 
H'(w  +  2K)=-  H'(w),     H'(w 

It  follows  that 

f(u  +  2 
We  further  have 

H'(tt   +2^0  771          H'(M) 

H(w  +  2iK;)  "        K       H(w)  ' 
so  that 

f(u  +  2iK')=v'f(u)-  vf7£ey». 

K. 

*  First  noted  by  Mittag-Leffler,  Comptes  Rendus,  t.  90,  p.  178;  see  also  Halphen, 
Fonct.  Ellipt.,  t.  I,  p.  232. 


270  THEORY   OF   ELLIPTIC    FUNCTIONS. 

The  function  f(u)  is  therefore  not  a  doubly  periodic  function  of  the 
second  sort.  It  will  nevertheless  serve  for  the  formation  of  a  doubly 
periodic  function  of  the  second  sort  with  the  factors  e2Ky  and  e2K'iy,  which 
function  becomes  infinite  on  an  arbitrary  number  of  points  within  the 
period-parallelogram. 

Let  F(u)  be  the  function  required,  so  that 

F(u  +  2K)  =  vF(u),      v 
and  F(u  +  2  iK')  =  v'F(u),     v' 

We  shall  express  F(u)  in  terms  of  f(u)  =       W  eyu. 


The  period-parallelogram  is  to  be  chosen  so  that  F(u)  does  not  become 
infinite  on  its  sides. 

We  again  form  the  function 


We  shall  see  that  ^(£)  is  here  not  a  doubly  periodic  function  of  the  first 
sort  as  was  the  case  in  Art.  234. 
From  the  formulas 


it  follows  that  -*!?(£  +  2  K)  = 

and  further  that 


^  .Rl)  _  ^  +    K._ 


We  again  note  that  2  iK'  is  not  a  period  of 

We  compute  next  2  Res  ^(£)  for  the  interior  of  the  parallelogram 
whose  sides  are  2  K  and  2  iK'  .  It  is  seen  that  $  =  u  is  an  infinity  of 
-^(£);  for  H(0)  =  0,  and  as  H(w)  is  an  odd  function,  its  expansion  is 

H(w)=  u(c0+ 
so  that  TT//,A       i 


where  P(u)  is  a  power  series  in  positive  integral  powers  of  u, 
Similarly  we  have 


Further,  since  e^  =  1  + 


where  P\(u  —  0  denotes  a  power  series  in  positive  integral  powers  of 


DOUBLY   PERIODIC   FUNCTIONS    (SECOND    SORT).         271 
The  expansion  of  F(£)  in  the  neighborhood  of  £  =  u   is 


We  therefore  have 


As  in  Art.  234, 


Res  ^(0  -  Aki  if(u  -  ak)  - 

'  * 


(u  -  ak) 


It  follows  that 

k  =  nr  4 

V  Res  +(*)  =  -F(u)  +  J)    Ak,if(u-ak)  -  ™*f  («  ~  «*) 


(4 -I)'/ 

We  cannot  put   S  Res  ^r(£)  =  0,  as  in  Art.  234;  but  after  Cauchy's 
Theorem 


where  the  integration  is  to  be  taken  over  the  four  sides  of  the  parallel 
ogram  in  the  figure. 


We  have  as  in  Art.  92 


or  bv  Art.  92, 


=  2K  (P  +  2Kt)dt  +  2iK' 


iK'+  2Kt)dt 


But  since  ^(£)  has  the  period  2  K,  it  follows  that 


2  7:1       Res  V(c)  -  2  K 


2iK't)dt 


or 


=  2K 


Res    r(£)=- 


+  2  /ft)  -  ^(p  +  2  tKr  +  2  KO    dt 

F(p  +  2  K«)  ^  ^  dt; 


re-(P+2K»  F(p  +  2  Kt)dt. 


272  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  definite  integral  is  a  quantity  independent  of  u,  which  we  may 
denote  by  A,  'so  that  therefore 


Equating  the  two  expressions  that  have  been  found  for  2  Res  ^(£),  it  is 
seen  that 


Further,  since  F(u  +  2  iK')  =  v'F(u),  we  may  write 

k  =  n  , 

F(u  +  2iK')  =  v'Aey»  +  i/  V     Ak,if(u  -  ak)- 


k=i 


(4-1)! 

On  the  other  hand  if  we  put  u  +  2  iK'  for  u  in  the  expression  above, 
we  have 


v'ni 


(4  -  1)! 
Comparing  the  two  results  just  derived,  it  is  seen  that 


This  condition  must  be  satisfied  by  the  A1  8  in  the  formation  of  the 
function  F(u). 

Since  7-  is  an  arbitrary  quantity,  it  may  be  made  equal  to  zero.  We 
then  have 


But  Akti  is  the  residue  of  F(£)  for  £  =  a&. 
We  therefore  have 

SAfc,i  =  SResF(f); 

and  consequently 

S  Res  F(£)  =  0,    when  7-  =  0. 


DOUBLY   PERIODIC   FUNCTIONS    (SECOND   SORT).         273 

But  if  f  =  0,  then  F(u  +  2  K)  =  F(u)  -    ' 

and  F(u  +  2iK')  =  F(u), 

so  that  F(u)  is  a  doubly  periodic  function  of  the  first  sort. 

We  thus  have  another  proof  of  the  theorem*  (see  Art.  99)  that  for  a 
doubly  periodic  function  of  the  first  sort  the  sum  of  the  residues  with 
respect  to  all  its  infinities  ivithin  a  period-parallelogram  is  equal  to  zero. 

ART.  238.  A  preliminary  formula  of  addition.^  —  By  means  of  the 
above  results,  and  as  an  illustration  of  them,  we  may  compute  the 
addition-theorem  for  sn  u. 

In  the  function  sn(u  +  v)  we  consider  v  as  constant  and  u  as  the  vari 
able.  This  function  becomes  infinite  on  the  points  where  &(u  +  v)  is 
zero,  viz., 

u  +  v  =  2  mK  +  (2  n  +  \)iKr. 

It  is  seen  that  Q(u  +  v)  vanishes  on  the  point  u  +  v  =  iK'  or  u  = 
iK'  —  v  and  on  all  congruent  points  (modd.  2  K,  2  iK'). 

It  is  quite  possible,  when  we  consider  the  parallelogram  of  periods, 
that  the  point  iK'  —  v  does  not  lie  within  it.  There  is,  however,  some 
congruent  point  which  does  lie  within  it,  and  we  shall  simply  denote 
this  point  by  iK'  —  v. 

Consider  the  product 

sn(u  +  v)  {  sn  u  —  sn  (iKf  —  v)}. 

If  u  =  iK  —  v,  the  expression  within  the  braces  becomes  zero  of  the 
first  order,  while  sn(u  +  v)  is  infinite  of.  the  first  order.     The  product 
therefore  remains  finite  for  u  =  iK'  —  v. 
We  form  next  the  function 

G(u)  =  sn(u  +  v}  {  snu  -  sn(iK'  -  v)  }\snu  -  sn  (iKf  +  2  K  -  v}  }  . 

This  product  remains  finite  for  u  =  iK'  —  v  and  for  u  =  iK'  +  2  K  —  v 
and  for  all  points  congruent  to  these  two  points  (modd.  2  K,  2  iKf). 
We  have 


sn(iK'-  v)  = 


sn  (—  v)  k  sn  v 

It  follows  that 

G(u)  =  sn(u  +  v}  \>sn  u  H  --  >  )  sn  u  —  —  ^  —  ', 
/  ksnv)  I  ksnv) 

=  sn(u  +  v)  \  sn2u  —  —  —  —  I  , 
(  k2sn2v  ) 

or  G(u)k2sn2v  =  sn(u  +  v}{k2sn2u  sn2v  —  1}  =  F(u),  say. 

*  See  Forsyth,  Theory  of  Functions,  p.  280. 

t  Hermite's  "  Cours  "  (Quatrieme  edition,  p.  242)  ;  see  also  Appell  et  Lacour,  Fonctions 
Elliptiques,  p.  129. 


274  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  follows  at  once  that 

F(u  +  2K)=-  F(u),   so  that    y  =  -  1 
and 

F(u  +  2iK')=  F(u),  or  i/^  1. 


We  note  that  F  (u)  is  a  doubly  periodic  function  of  the  second  sort  with 
the  periods  2  K  and  2  iK' .  Consider  the  parallelogram  with  the  sides 
2  K  and  2  iK'  in  which  the  point  iK'  lies.  The  function  F(u)  becomes 
infinite  on  this  point  but  on  no  other  point  of  the  parallelogram. 

To  determine  the  order  of  the  infinity  of  F(u)  for  the  point  u  =  iK', 
it  is  seen  that 

sn  h  =  h  +  c3h3  -f  c5h5  +  -   •   •   ; 

and  consequently  if  we  put 

u  =  iK'  +  h     or     h  =  u  —  iK', 
we  have 

sn(iK'+h)=      1  l 


k  sn  h      kh     1  +  e3h2  + 

-—  f  1 
kh* 

It  follows  at  once  that 

and  consequently 

k2sn2usn2v  —  1  = -sn2v  +  2e2sn2v  +  •  •  •  —  1. 

(u  —  iK'}2 

Noting  that =  en  v  dn  v,  it  is  seen  that  the  expansion  of  sn(u  +  v) 

in  the  neighborhood  of  u  =  h  +  iK'  is 

1 


sn(u  +  v)  =  sn(v  +  iK'  +  h)  = 


(u  - 


k  sn  (v  +  h) 
which  by  Taylor's  Theorem 

1        _  7  en  v  dn  v   , 
k  sn  v  k  sn2v 

1       _  cnvdnv 
k  sn  v        k  sn2v 
We  therefore  have 

•n,  \      1         sn  v  en  v  dn  v        1          ,    n ,         . r,/x 

F(u)=k(u-iK')*-     —-^^  +  P(U-*K)- 
Writing 

_  en  v  dn  v  ^    *         snv  _    * 

IV  IV 

we  have 


DOUBLY   PERIODIC   FUNCTIONS    (SECOND   SORT).         275 

We  shall  next  express  F(u)  through  a  fundamental  function  f(u). 
The  function  f(u)  must  be  a  doubly  periodic  function  of  the  second  sort 
with  the  factors  +  1  and  -  1  and  with  the  periods  2  K  and  2  iK'. 

We  may  consequently  choose  -  for  this  fundamental  function.  We 
have 

-  =  —  h  positive  powers  of  u. 
snu      u 

Consequently  we  have  Res/(w)=  1  =  C  (of  Art.  234). 

u  =  0 

Hence  (see  the  formula  at  the  end  of  Art.  234)  it  follows  that 
F(u)  =  A0f(u  -  iK')-  Ai/'(t*  -  iK'). 

We  have  further      ff          .v,^  1  , 

f(u  -  iK')  =  —  --  —  —  =  k  sn  u, 
sn(u  —  iK) 

and  also  f'(u  —  iK')  =  kcnu  dn  u,  so  that 

rv  \          cnv  dnv  j  snv  -, 

F(u)  =  --  -  -  k  snu  --  k  en  u  dn  u. 

K  K 

Equating  the  two  values  of  F(u),  it  is  seen  that 

sn(u  +  v)  [k2sn2u  sn2v  —  1]  =  —  sn  u  en  v  dn  v  —  sn  v  en  u  dn  u, 
or  finally  ,  >.      sn  u  en  v  dn  v  +  sn  v  en  u  dn  u 

87l(U  -h  V)  — 


1  —  K-sn^u  sn^v 

which  is  the  addition-theorem  for  the  modular  sine. 

When    k  =  0,   we    have    sn  u  =  sin  u,   en  u  =  cos  u,    dn  u  =  1,    and 
consequently 

sin  (u  +  v)  =  sin  u  cos  v  +  cos  u  sin  v. 

The  above  addition-theorem  may  also  be  written  in  the  form 


«,<„+„)-_     dv          du 


1  —  k2sn2u  sn2v 

As  an  exercise  the  student  may  derive  the  addition-theorems  for 
cn(u  +  v)  and  dn(u  +  v)  and  compare  the  result  with  those  given  in 
Chapter  XVI. 

ART.  239.  As  a  further  application  of  the  doubly  periodic  functions 
of  the  second  sort  we  may  develop  in  series  of  sines  and  cosines  such 
expressions  as 

Q(u  +  a)         H(^  +  a)        QI(U  +  a)         fii(u  +  a) 


which  appear  in  Jacobi's  investigations  relative  to  the  rotation  of  a  body 
which  is  not  subjected  to  an  accelerating  force.* 

*  Jacobi,  Werke,  II,  pp.  292  et  seq. 


276  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Consider  with  Hermite  *  the  series 


e  K 


sin  —  (u  + 
2  K 

where  n  takes  all  values  from  —  oo  to  +  oo ,  a  being  a  constant  which 
will  be  represented  by  a  +  ia'. 

We  shall  first  show  that  this  series  is  convergent,  whatever  be  the  value 
of  u,  provided  that  a'  is  less  in  absolute  value  than  2  K' . 

Writing  the  general  term  in  the  form 


. 

eK  -  e 


it  is  seen  that  we  may  neglect  the  first  or  the  second  exponential  term 
in  the  denominator  according  as  n  becomes  positively  or  negatively 
indefinitely  large. 
We  thus  have  either 


-e  or         e  . 

If  we  write  —  n  in  the  place  of  n  in  the  second  of  these  quantities  and 
take  the  limit  for  n  indefinitely  large  of  the  nth  root  of  the  moduli,  we 
have  after  a  has  been  replaced  by  a  +  ia' 


either      e    -  or     e 

If  for  the  first  a'  +  2  K'  >  0  and  for  the  second  a'  —  2  K  <  0,  the  two 
limits  are  less  than  unity  and  the  series  in  question  is  convergent. 
Consider  next  the  function 


sin  —  (u  +  2  niK') 
2  K 

and  noting  that,  since  n  varies  from   —  oo  to   +  oo ,  we   may  change  n 
into  n  +  1,  we  have 

(n-\-\}ifia  .  itina 

K  ^  rit 


Sin  JL.  [u  +  2  (n  +  l)iK']  sin  --  [u  +  2  iK'  +  2  niK'] 

2  K  2  K 


It  follows  at  once  that 


or 


*  Hermite,  Ann.  de  I'Ecole  Norm.  Super.,  3e  se>ie,  t.  II  (1885);  see  also  Hermite, 
Sur  quelques  applications  desfonctions  elliptiques,  p.  35. 


DOUBLY   PERIODIC   FUNCTIONS  (SECOND   SORT).         277 
On  the  other  hand  we  have  immediately 


so  that  $(u)  is  a  doubly  periodic  function  of  the  second  sort  with  the 

tVa 

multipliers  —  1  and  e    K  . 

The  poles  are  obtained  by  writing 

sin  ^-  (u  +  2  niK')  =  0, 

from  which  we  have 

u  =  2  mK  -  2  niK', 

where  m  is  an  arbitrary  integer. 

We  therefore  see  that  on  the  interior  of  the  rectangle  of  periods  2  K 
and  2  iK'  there  is  only  one  pole  u  =  0,  the  corresponding  residue  being 

9  jr 

-  --     We  further  note  that  the  quantity 

2K  H' 


n         H(u)9(a) 

has  the  same  multipliers,  the  same  pole,  and  the  same  residue. 
We  may  therefore  write  (see  Art.  83) 

idna 

2K  H'(0)0(M  +  a)  = 
-         H(w)0(a) 

11  2^ 

If  a  and  u  are  permuted  in  this  equation,  we  have 
(1v  2K  ^(0)0(1^  +  a)  _^ e^ 


We  may  deduce  the  others  as  follows: 
If  we  change  a  into  a  +  iK',  we  have 


2K 


or 

(2n  +  l)siu 


u  +  a) 


g 

2  A. 


278  THEORY   OF   ELLIPTIC    FUNCTIONS. 

If  further  a  +  K  is  written  for  a  in  (1)  and  (2),  these  formulas  become 

Ttinu 

2K  ~ 


(3) 


2 


a)  _      •  -     2« 


~-^[«*<2« 

If  u  +  K  is  written  for  u  in  the  four  formulas  above,  we  have  the  four 
following  formulas,  in  which  ®i(u)  is  found  in  the  denominators: 

Ttinu 

tK\  2K  H'CO)©!^  +  a)      ^         (-  l)ne^~ 

W  ^     /.ATJ/.N  ~2^~   ~~^~       

sin (a  +  2  niK') 

2  A. 

(2n  +  l)atu 
;2n  +  l^        2K 


ART.  240.     Hermite  next  formed  a  series  entirely  different  from  the 
one  of  the  preceding  Article  which  is  represented  as  follows: 


where  n  takes  all  even  integral  values  from  —  oo  to  +00,  while  the  quan 
tity  e  must  be  supposed  zero  for  n  =  0  and  equal  to  unity  positive  or 
negative  according  as  n  is  positive  or  negative. 
If  we  allow  n  to  take  only  the  positive  integers 

n  =  2,  4,  6,   -   -   •  , 
the  series  above  may  be  decomposed  into  the  two  partial  series 

jdna 

cot  ||  +  cot  ||  +  %e  2X  [cot  ^  («  +  niK')  +  t] 


DOUBLY   PERIODIC    FUXCTIOXS  (SECOND   SORT).         279 
which  by  an  easy  transformation  becomes 


(na+nitf  +u) 


cos  -^7  (u  +  niK') 
2  A 

-—.(na-niK'  +  u) 

cos  — —  (u  —  niK') 

To  prove  the  convergence  of  this  series,  note  that  for  large  values  of  n 
the  two  denominators 

cos  -^  (u  +  niK'}    and     cos  —  (u  -  niK') 
£  A  2  A 

may  be  replaced  by 

\e    2X  and     ^  e2  K 

the  general  terms  becoming 


If  we  put  a  =  a  +  ta',  we  have  for  the  limit  of  the  nth  root  of  their 
moduli  as  n  becomes  very  large,  the  quantities 


i         2 

e  and    e2A 

and  consequently  the  conditions 

a'+  2K'>0,  a'-  2  K'  <0. 

It  follows  that  the  series  in  this  Article  is,  as  the  one  in  the  preceding 
Article,  convergent  when  the  coefficient  of  i  in  the  constant  a  is  in  abso 
lute  value  less  than  2  K'  '.  This  series  also  defines  a  doubly  periodic 
function  of  the  second  sort.  For  writing 


¥(11)  =  cot  —^ 
we  have  the  relations 

-ia 

The   second  of   these   relations  is  evident   from   the   expression   of   the 
product  e  K  *&(u  +  2  iK'),  viz., 

T\GL  ;rtct  Trtfl  T^ino. 

~jr          -jid          ~jr  ^-\     OK  r             7T 
^6       COl r€         76"      I  COt 

2  A  ^  2  A 


280  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

We  have 


and  if  we  change,  as  is  permissible,  n  to  n  —  2  in  the  general  term,  it 
becomes 


-. 

\  cot-2-  (u  +  niK')  +  ei  \  , 
L       2  K  J 


2  K. 
where  now  there  is  a  modification  regarding  £. 

The  quantity  e  must  be  =  1  for  n  =  4,  6,  8,  .   .   .  ,  while  e  =  0  for 
n  =  2  and  e  =  —  1  for  n  =  0,  —  2,  —  4,   •  •   •  .    We  note  that  in  adding 

?rta 

to  the  terms  corresponding  to  n  =  2  and  n  =  0  on  the  one  hand  ie  K  and 

/    Trta  \ 

on  the  other  i,  and  consequently  in  causing  the  quantity  i\eK  +  I/  to 
enter  the  summation,  we  find  for  e  precisely  the  significance  which 
was  accorded  it  in  the  function  W(u).  We  further  note  that  within  the 
rectangle  of  periods  there  exists  the  one  pole  u  =  0,  to  which  corresponds 

9   /<" 

the  residue  -  .     We  may  therefore  represent  the  function  W(u)  by 

2K  H'0Htt  +  o 


If  we  interchange  u  and  a  we  have  finally 


where  n  represents  all  even  integers  and  the  unity  e  must  be  taken 
positive  when  n  is  positive  and  negative  when  n  is  negative.  *iu 

Next  changing  a  to  a  +  iHC',  we  have,  after  having  multiplied  by  e2K, 
the  formula 


where  m  denotes  the  odd  integer  n  +  1. 
Since 


we  have,  if  the  term  ie2Kis  introduced  under  the  summation  sign, 


where  m  represents  all  odd  integers  and  s  must  be  taken  +  1  or   —  1 
according  as  m  is  positive  or  negative.     Changing  a  to  a  +  K  we  have 


DOUBLY   PERIODIC   FUNCTIONS  (SECOND   SORT).         281 

the  formulas  (11)  and  (12)  below,  and  by  replacing  u  by  u  +  K  in  the 
formulas  (9),  (10),   (11),   (12)  we  have  the  formulas  below,   (13)     (14) 
(15),  (16). 


2  K  H' 


no, 

IT 


/1/n 


Hl(M)e(a) 


(16) 

n 

the  quantities  m,  n,  and  e  being  denned  as  above. 

EXAMPLES 

1.    If  w  =  1,  3,  5,  .  .  .    ;  n  =  2,  4,  6,   .   .   .  ,  show  that 
2tf 


0(M)H(a) 

2.    Further  if  mf  =  1,  3   5,  .  .  .  ,  prove  that 


0(n)0(a) 
3.    Show  that 


.      HfttfHtM          ^2K        -        L^1^' 
i;       H(,)0i(a)a   =cosec^-^2A[tan^(^^^)-^]- 


0(M)Hl(a)  ^  -  — 

4.    Prove  that 


cos— 


[Kronecker.] 


CHAPTER  XIII 
ELLIPTIC  INTEGRALS  OF  THE  SECOND  KIND 

ARTICLE  241.  From  the  investigations  relative  to  the  integrals  of 
the  first  kind  in  Legendre's  normal  form  (see  Chapter  VII)  it  is  seen 
that  the  elliptic  integral  of  the  second  kind 

z2dz 

-  z2)(l  -k2z2) 

is  finite  and  continuous  on  the  finite  portion  of  the  Riemann  surface. 
In  the  neighborhood  of  the  point  z  =  oo,  we  have 


-  z2)(l  -  & 
so  that 


-  z  +  £i  _L  ^2     ,     .     .    . 

—    ^    T^  1^        v      I  ) 


-  z2)(l  -  &2z2)  z       z3 


where  the  a's  and  6's  are  constants. 

It  follows  that  the  elliptic  integral  of  the  second  kind  is  algebraically 
infinite  of  the  first  order  for  the  value  z  =  oo  in  both  the  upper  and  the 
lower  leaves. 

In  the  Weierstrassian  normal  form 

*£ 

'W 

the  expansion  in  the  neighborhood  of  the  point  Z  =  oo,  which  is  a  branch 
point,  is 


the  limits  of  integration  being  so  chosen  that  no  constant  term  appears 
in  this  development.  The  question  naturally  arises  whether  it  is  possible 
to  form  a  one-valued  function  of  position  on  the  Riemann  surface  which 
is  algebraically  infinite  at  only  one  point. 

To  investigate  this  question,  consider  the  integral 

Cdt 


where  C  is  a  constant. 

282 


ELLIPTIC    INTEGKALS   OF   THE   SECOND   KIND.          283 

This  is  the  simplest  integral  which  is  algebraically  infinite  of  the  first 
order  at  the  two  points* 

a,  VS(a)     and     a, -VS(a). 

We  note  also  that  the  integral 

At  4-  B 


f 


(t  -  a 


dt, 


where  A  and  B  are  constants,  becomes  infinite  in  the  same  manner  at  the 
same  two  points  as  the  integral  above.  Neither  of  these  integrals  is 
infinite  for  t  =  oo. 

We  shall  so  choose  the  constants  A  and  B  that  the  latter  integral  be 
comes  infinite  on  the  point  a,—  \/S(a)  in  the  same  manner  as  does  the 
first  integral. 

By  Taylor's  Theorem  we  have  in  the  neighborhood  of  the  point  t  =  a 


At+JS  =  Aa  +  B  +  _  _  (f       ^  | 

VS(t)         VS(a)  S(a) 

It  follows,  if  we  put 

(1)  AVSM  -  I  (Aa  +  B)  -^ZL  =  J  =  0, 

2  VS(a) 

that 


(At  +  B)dt         _        Aa  +  B  log  (t  -  a) 


r    (At  + 

J    t-  a2 


(t-  a)2VS(t)  (t  -  a)  VS(a)  S(a] 


p(t  _    . 


will  not  contain  a  logarithmic  term  in  the  expansion  according  to  ascend 
ing  powers  of  t  —  a. 
Further,  since 

Cdt  C 


(t  -  a)2  t  -  a 

it  is  seen  that  the  two  integrals  become  infinite  alike  on  the  point 

a,-VS(a),  if 

(2)  -a 


VS(a) 
It  follows  from  equations  (1)  and  (2)  that 

1  =       1        S'(a)  ^ 
2 


B=-C     S       -  Aa  =  C 


VS(a) 


*  The  following  results  are  true  not  only  when  S(t)  is  of  the  third  degree  in  t,  but 
also  when  this  degree  is  n,  where  n  is  any  positive  integer. 


284  THEORY   OF   ELLIPTIC    FUNCTIONS, 

and  consequently  that  the  integral 


At  +  B 


-\dt 


(t-a)2       (t-a)2VS(t)/ 

dt 


J 


is  an  integral  of  the  second  kind,  which  is  infinite  of  the  first  order*  at  only 
the  one  position  (a,  \/S(a)).     Write  C  =  %  and  put 


_  a) 


2(t-  a)2 


We  may  regard  this  integral  as  the  fundamental  integral  of  the  second 
kind. 

ART.  242.  We  next  raise  the  question:  Is  there  another  integral 
EI($,  VjS(p)  of  the  second  kind  which  becomes  algebraically  infinite  of 
the  first  order  on  the  point  a,  \/$(a)?  If  such  an  integral  exists,  its 
development  in  the  neighborhood  of  t  =  a  is  of  the  form 

-  a). 


t  —  a 


Writing  E  i  (t,  VS(f))  =  E  (t,  VS(T)  )  , 

it  is  seen  that 


E(t,VS(t))-E0(t,VS(t)) 

does  not  become  infinite  for  any  point  on  the  Riemann  surface.  It  is 
therefore  an  integral  of  the  first  kind,  =  F(t,\/S(t)),  say.  It  follows 
that 

E(t,  VS(ij)  -  EoO,  V^)  +  F(t,  VSW>). 

Hence,  if  we  add  to  an  integral  of  the  second  kind  an  integral  of  the  first 
kind,  we  have  an  integral  of  the  second  kind  which  is  infinite  only  at  the 
point  (a,  VR(a)  )  provided  the  original  integral  of  the  second  kind  is  infinite 
only  at  this  point.  There  are  consequently  an  infinite  number  of  inte 
grals  of  the  second  kind  which  are  algebraically  infinite  of  the  first  order 
on  the  one  point  (a, 


*  Cf.  Koenigsberger,  Elliptische  Functionen,  p.  250. 


ELLIPTIC   INTEGRALS   OF   THE    SECOND   KIND.          285 

ART.  243.     If  we  put 

(t  -  a) 


and 
then 


da          2(t  -  a)2 
We  further  write  (see  Art,  287) 


which  integral,  as  we  saw  above,  becomes  algebraically  infinite  at  infinity. 
It  is  then  evident  that  the  expression 


remains  finite  and  continuous  in  both  the  finite  and  the  infinite  portion 
of  the  Riemann  surface.  It  is  therefore  an  integral  of  the  first  kind. 

Similar  results  hold  when  mutatis  mutandis  S(t)  is  of  the  fourth  degree 
in  t.  It  is  thus  seen  that  the  elliptic  integral  of  the  second  kind,  which 
becomes  algebraically  infinite  at  the  point  infinity,  may  be  replaced 
by  one  which  is  algebraically  infinite  at  only  one  position  on  the  Rie 
mann  surface,  the  latter  position  being  a  definitely  prescribed  one. 

ART.  244.     If  in  the  integral  of  the  first  kind 


fzs 

u  =   I      — = 

«Au  v(i 


dz 


we  put  z  =  sin  </>,  we  have  in  Legendre's  notation 

Ffak)  =  r  W    ==  P4^-»      where    A0  =  Vl  -  k2  sin26- 

Jo  Vl-A;2sin20     Jo   A^> 

The  complete  integrals  of  the  first  kind  are  therefore 


Js 

Vl  -T'2  sin2<£ 

In  Legendre's  notation  (Fonct.  Elliptiques,  t.  I,  p.  15)  the  integral  of 
the  second  kind  is 


E(k,<f>)=  fVl  -  k2sm2<i>d<f>  = 

J  o  «/o 


286  THEOEY   OF   ELLIPTIC   FUNCTIONS 

The  complete  integrals  are  (see  also  Art.  249)  : 


=  E, 


E 

JO      \/l   _  Z2  o       YI   _  Z2 

If  we  put  d<j)  =  dnu  du,  A^>  =  dnu,  we  have 

E(k,<t>)=  E(u)=  CUdn2udu  -  P(l  - 
Jo  Jo 

(Jacobi,  Werke,  I,  p.  299.) 
ART.  245.     To  study  the  integral  of  the  second  kind 


r 
«/o 


-  z2)(l  - 
as  a  function  of  u,  where 

dz 


r 

Jo,i 


u  =  

-  z2)(l  - 

we  may  with  Hermite  *  multiply  this  integral  by  k2  and  put 

'u)  =   I    k2sn2u  du. 


f* 

I 

Jo 


We  note  that  the  function  sn2u  has  the  periods  2K  and  2iK'',  and 
from  the  developments  above  it  is  seen  that  /2(^)  is  a  one-valued  function 
of  z.  But  z  considered  as  a  function  of  72  is  not  one-valued,  and  con 
sequently  the  problem  of  inversion  for  these  integrals,  which  is  effected 
with  difficulty,  does  not  lead  to  unique  results  (see  Casorati,  Ada  Math., 
Bd.  8). 

ART.  246.     We  saw  (Art.  217)  that  snu  became  infinite  on  the  points 

2mK  +(2n  +  l)iK'  =  a,    say. 

Writing  u  —  a  =  h  or  u  =  a  +  h,  we  must  develop  sn2u  =  sn2  [2  mK 
+  (2  n  +  l)iK'  +  h]  in  powers  of  h.  Since  sn2u  has  the  periods  2  K  and 
2  iK',  we  have 

1 


sn2  [2mK  +(2n  +  I)iK'  +  h]  =  sn2  (<*#'  +  h) 


k2sn2h 


*  Hermite,  Serret's  Calcul,  t.  II,  p.  828;  CEuvres,  II,  p.  195;  Crelle's  Journ.,  Bd.  84. 
This  integral  Hermite  denotes  by  Z(u).  We  shall,  however,  reserve  this  symbol  for 
the  integral  employed  by  Jacobi  (Art.  250) . 


ELLIPTIC   INTEGKALS    OF   THE   SECOND   KIND.          287 

so  that 


,  or  k2sn2u 


, 
k2sn2h  sn2h      h2  +  ch4  +  • 

It  follows  that 

k2sn2u  =  -  -  ?—  -  4-  e0+  CI(M  - 
(u  -  a)2 

and  consequently,  since  the  integrand  does  not  contain  the  term  (u—a)~l, 
the  integral 

/2(M)  _   CUk2sn2u  du 
Jo 

is  a  one-valued  function  of  u. 

ART.  247.  The  analytic  expression  for  I2(u).  —  The  function  k2sn2u  is 
doubly  periodic  of  the  first  sort,  having  the  periods  2  K  and  2  iK'.  The 
only  infinity  within  the  period-parallelogram  having  the  sides  2  K  and 
2  iK'  is  iK'. 

We  may,  however,  consider  k2sn2u  as  a  doubly  -periodic  function  of  the 
second  sort  with  the  factors  v  =  1  and  i/=  1;  or  v  =  e2^,  i/  = 
where  y  =  0. 

We  have  here  the  exceptional  case  of  Art.  237  where 

F(u)  =  Ce^  +  2)  U»,i/[t«  -  a)  -  %V'("  -« 


the  function  F(u)  being  fc2sn2M  and  f(u)  =  e>"  being  /(M) 

_  tlvti] 

since  7-  =  0. 

The  development  of  k2sn2u  in  the  neighborhood  of  the  infinity  iK'  is 


Hence  in  the  formula  above,  Ak,i,  the  coefficient  of  (u  —  tK')-1,  is 

zero:  and  A^  2,  the  coefficient  of  —  (u  —  iK')'1,  is  —  1. 

du 

We  consequently  have 

k2sn2u  =  C-//(^^  -  iK'). 
It  follows  that 


k2sn2u  du=[Cu  -  f(u  -  iK')] 

_ 


r/,      r        TL'(u-iK' 

/2(w)  =  Cu  ~ 


It  is  thus  seen  again  that  I2(u)  is  a  one-valued  function  of  u. 


288  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Since 


we  have 

ic  K'       idu 

H  (u  -  iK')  =  -iel*+™:®  (u) 

-jjf( 

=  -ie    4X 
It  follows  that 


We  therefore  have 

W(u  -  iK')  m  jn_ 
K(u-iKf)       2K       0(w) 

and  W(-  iK')  =_  jn_      er(0) 

H(-{Kr)       2K      6(0) 
since  ©'(0)  =  0. 

It  has  thus  been  shown  that* 


To  determine  C,  we  have  from  above 


Equating  powers  of  u  on  either  side  of  this  equation,  we  have 

r  _Q"(0) 

=  w 

It  follows  that 


fUI2(u)du  =  %Cu2-  Iog0(w)  +  C', 

*J  Q 


where  C'  is  the  constant  of  integration. 
From  this  it  is  seen  that 


'0 

or 


Finally  we  may  write  f 

0(tt)=C"'eiC"'--C7'(u)du,     where    C"=  0(0). 

*  Hermite,  Serret's  Calcul,  t.  2,  p.  829. 

t  Jacob!  (Crelle,  Bd.  26,  pp.  86-88;  Werke,  II,  pp.  161-170)  defines  the  0-function 
by  this  formula  and  therefrom  derives  directly  the  series  through  which  this  tran 
scendent  may  be  expressed  and  its  other  characteristic  properties. 


ELLIPTIC   INTEGRALS   OF    THE    SECOND   KIND.          289 
ART.  248.     We  may  next  consider  the  integral  of  the  second  kind 


0,1  V(l  -  z2)(l  -  k2z2) 

regarded  as  a  function  of  z,  s  on  its  associated  Riemann  surface. 

In  the  simply  connected  Riemann  surface  T',  we  saw  that  u(z,  s)  was 
a  one- valued  function  of  z,  s.  If  z,  s  are  given,  then  Ti(z,  s)  is  uniquely 
determined,  and  if  u  is  known,  then  also  /2(w)  is  known.  Hence  in  T' 
not  only  the  elliptic  integral  of  the  first  kind  but  also  the  elliptic  integral 
of  the  second  kind  is  a  one- valued  function  of  z,  s.  Since  /jt(z,  s),  that 
is,  the  elliptic  integral  of  the  second  kind  in  I",  is  a  one-valued  function 
of  z,  s,  it  is  independent  of  the  path  of  integration.  This,  however,  is  not 
true  of  /2(z,  s),  that  is,  of  the  integral  of  the  second  kind  in  the  Riemann 
surface  T  which  does  not  contain  the  canals  a  and  b. 

For  the  elliptic  integral  of  the  first  kind  77(z,  s)  we  had 


(  ""  M  -  u(p)  =  A  (k)  =  2  iK'  on  the  canal  «, 
I  u(p)  -  u(X)  =  B(k)  =  4  K  on  the  canal  b. 

In  a  corresponding  manner  we  shall  represent  the  constant  differences  of 
the  integral  of  the  second  kind  at  opposite  points  of  the  banks  as  follows:  * 

(  /2(/)  —  /2(/o)  =  2iJ'  on  the  canal  a, 
( Iz(p)  —  /2(/)  =  4  J  on  the  canal  b. 


We  had  (Art.  193) 


K'  = 


r1 dz 

Jo   Vn  -  z2)(l  -  k2z2] 


:>-  /"  ,       rfz          -  f 

Jo  V(l  -  z2)  (I  -  k'2z2}      Ji 


1 

dt 


)      Ji  V(t2  -  1)(1  -  k2t2) 


In  a  corresponding  manner  we  may  write  with  Weierstrass  (Werke,  I, 
pp.  117,  118) 

i 

A;2*2^ 


=  r1 

Jo 


-  z2)(l  -  A;2z2) 


Jf 

' 


We  note  that  J'  is  not  deduced  from  J  by  changing  k  to  k'. 

From  these  definitions  of  J  and  J',  it  is  seen  in  the  remark  at  the  end  of 
Art.  249  that  the  formulas  (2)  above  follow. 

*  Hermite,  loc.  cit.,  p.  828;  Fuchs,  Crelle,  Bd.  83,  pp.  13-38. 


290  THEORY   OF   ELLIPTIC   FUNCTIONS. 

AKT.  249.     We  had  above 


If  in  this  formula  we  write  u  =  K,  we  have 


From  the  formulas 

0(t*  +  £)=©i(tO,     &r(u  +  K)=Qlf(u), 

it  is  seen  that  for  u  =  0 

O'(K)  =©!'(())  =0, 
and  consequently  I2(K)=CK. 

To  compute  72CK)  we  put  u  =  K  in  z  =  snu,  and  if  z0  is  the  value  of 
z  that  corresponds  to  w  =  K,  we  have 


z0=  snK  =  1  (Art.  218). 
It  follows  that 

k2z2dz 


r 

t'O.l 


=  J. 


V  (1  -  z2)(l  -  k2z2) 
We  therefore  have 

J  =  CK,     or     C  =  j£; 

and  finally  j          @r(  } 

/2W-  —  ^  -  777^' 

K        8(v) 

We  may  next  compute  the  constant  C  in  a  different  manner.     If  in  the 
equation 


we  write  K  +  ^'K7  for  u,  it  becomes 


h(K  +  iK')  =  C(K  +  iX' 


2K 
To  compute  72(K  +  iK')  we  put  w  =  K  +  t'K7  in  sn  u. 

If  Zx  is  the  corresponding  value  of  z,  we  have  ~z.\  =  —.     Further,  since 

k 
i  i 

k 


Z2)(l 

-/ 

«/  0,1 


-  z2)(l  - 


ELLIPTIC   INTEGRALS   OF   THE   SECOND   KIND.          291 

we  have 

iJ'=I2(K+iK')-J, 

or  I2(K  +  iK')=J  +  iJ'; 

and  consequently 

J  +  iJ'=  C(K  +iKf)  +  ~ 

2  A. 

Eliminating  C  from  this  formula  and  the  formula  CK  =  J,  it  is  seen 
that 

J'K  -  K'J  =  |- 

We  note  that  _ 

r1  v/1  _  1^-7-2 

K-J=\    V  X      k  z  dz  =  E  (Legendre); 
•/fl     Vl  -  z2 

and  making  the  transformation 


C1\-/-\  Z*'2,,,2 

/  /       VI    -^      M 


it  is  seen  that 


\  1  -  u 
It  follows  that 


which  is  the  celebrated  formula  of  Legendre  (Fonct.  Ellipt.,  I,  p.  60). 

Remark.  —  The  characteristic  properties  of  /2(w)  are  expressed  through 
the  formulas 


These  formulas  follow  at  once,  when  we  note  that 


Change  u  to  u  +  2  K  and  z^  +  2  i'Kr  respectively  hi  the  equation 


and  use  the  relation  j^j,  _  jg/  =  £. 

ART.  250.     We  note  that 


J  f«         9     ,        0r(w) 

-u-    /     k2sn2udu  =  — ^: 
K         Jo  B(u) 


or 


292  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

With  Jacob!  (Werke,   I,   p.   189)  we  define  the  zeta-function  by  the 
relation 


u)=  (l  -  | ]u  -   rk2sn2u  du, 


which. is  Jacobi's  elliptic  integral  of  the  second  kind.     It  follows  *  also  that 
0 (w)=0 (0)/«  Z(u)du,     where     0(0)  =  \/^^-    (Art.  341). 

*          7T 

The  ©-function  may  thus  be  considered  as  originating  from  the  function 
7i(u)  [see  Cayley,  Elliptic  Functions,  p.  143]. 
From  the  formula 


we  have  dn2w  =  —  +  7t'(u)  and  consequently  Z'(0)  =  1  --- 
A.  K 

It  follows  at  once  that 

k2sn2u  =  Zr(0)-  Zr(^), 
and  k2cn2u  =  k2  -  Zr(0)  +  Z'(w)  ;   Zr(K)  =  Z'(0)  -  fc2 

It  is  further  seen,  since 


that 


As  ®i(tO  is  an  even  function,  its  derivative  is  odd,  so  that 

Z(K)=  0. 

ART.  251.  With  Jacobi  (Fund.  Nova,  §  56;  Werke,  I,  p.  214)  we  shall 
derive  other  properties  of  the  Z-function  and  at  the  same  time  we  may 
note  the  connection  with  the  ©-function.  We  emphasize  the  following 
results  because  the  properties  of  the  ©-function  are  again  derived  inde 
pendently  and  at  the  same  time  we  have  an  a  priori  insight  into  the 
Weierstrassian  functions.  In  Art.  220  we  made  the  imaginary  substitution 


It  follows  at  once  that 


*  Jacobi,  Werke,  I,  pp.  198,  224,  226,  231. 


ELLIPTIC   INTEGRALS    OF   THE    SECOND   KIND.          293 

This  expression,  when  integrated,  becomes 


or 

(1) 

It  follows  that 
(2) 

L 

From  the  formula  (Art.  249) 

FE(k')  +  F(k')E  -  FF(k')  =  ^ 

2i 

we  have  at  once 

')  =  -£-  [F(k')E(+,  *')  ~  E(V)FW,  k')] 
r  (K  ) 


'  2F(k') 
Equation  (2)  becomes  through  this  substitution 

,  k) 


> 
iF  F(kf) 


2  FF(k') 
Using  the  Jacobi  notation 

$  =  am  iu,     ^  =  am  (M,  fcr),     F(<£)  -  tu,     F(^,  A;')  =  u, 
we  have 


_  Z(     ,,,. 

and  consequently  from  (3)  we  have 

(4)  iZ(iu,  k)=-  tn(u,  k')  dn(u,  k')  +  -p-  +  Z(w,  A;'). 

Z  A  A 

Multiplying  (4)  by  dw  and  integrating,  this  equation  becomes 

fUiZ(iu,  k)du  =  log  cn(u,  k')  +     ~f    +  |    Z(u,  ^) 
Jo  4AA        Jo 

Further,  since 


it  follows  that 

(ef.Art.204). 


294  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Formulas  (4)  and  (5)  reduce  the  functions  Z(iu)  and  ®(iu)  to  real  argu 
ments. 

If  in  (5)  we  change  u  into  u  +  2  K',  that  formula  becomes 


In  this  formula  change  iu  to  u  and  we  have 

ir(K'  -  iu) 

(6)  ®(u  +  2  iK')  =  -e      K      ®(u)          (cf.  Art.  202). 

Again  write  u  +  K'  for  u  in  (5)  and  note  that 

cn(u  +  K',  kf)=-  k  sn ^  k"> , 
dn(u,k') 

Gi(»,    i    IT i  jpf\ dn(u,  k  )  s\,     -.,% 

Vk 

It  follows  that 

/-v  /  .  7r(M  +  A"')2 


.    @(0) 

w(2  u  +  A^r 

=  —  e      4K 


, 

0(0) 

Write  iu  for  w  in  this  formula  and  it  is  seen  that 

7r(K'  -  2  iu) 

(7)  ®(u  +  iK')=ie      4*      Vksnu®(u), 

which  is  a  verification  of  formulas  (V),  Art.  202,  and  (VIII),  Art.  217. 
By  taking  the  logarithmic  derivatives  of  (6)  and  (7),  we  have 

(8)  Z(u  +  2iK')=-^  +  Z(u), 

K 

(9)  Z(u  +  iK')=-  ^-  +  cotnudnu  +  Z(u). 

2  K. 

Write  u  =  0  in  formulas  (6),  (7),  (8),  (9)  and  we  have 

nK' 

0  (2  iK')  =  -  e  K  0  (0) ,     0  (iK')  -  0     (cf .  Art.  203), 

H, 
K. 


ELLIPTIC    INTEGRALS   OF   THE    SECOND   KIND.          295 
ART.  252.     In  Art.  227  we  saw  that 
=  const.  - 


1  -  q-        2(1-  q4) 
q3  cos  6  u  _    q4  cos  8  u 
3(l-g«)      4  (1  -  f)  - 
From  the  relation 


it  follows  that 


-  q2m 


[Jacobi,  Werke,  I,  p.  187.] 
We  also  have 

(2)  Z(tt)~e(to~i3: 

"  K 

To  be  noted  is  the  equality  of  the  right-hand  sides  of  (1)  and  (2).     We 
further  note  that 


2Ku  -JK_  [£cos2M   ,   2 g2  cos 4 ?* 

=  - 


ART.  253.     Thomae  *  introduced  the  notation 


Differentiate  logarithmically 

and  we  have  2 .2.  „ 

rj     t,.\      rj  f..\         k^snucnu 

Similarly  we  have 


dn  u 
[Jacobi,  W^erke,  I,  p.  188,  formula  (6).] 

/  x  _  en  u  dn  u 

sn  u 

/  v  =  _  sn  u  dn  u 
en  u 

*  Thomae,  Functionen  einer  complexen  Veranderlichen,  pp.  123  et  seq.;  Sammlung 
von  Formeln,  etc.,  p.  15. 


296  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  254.     The  derivatives  of  the  Z-functions  are  one-valued  doubly 
periodic  functions;  for  differentiating 


K 

it  is  seen  that 

J_ 

K 

Further,  since 

it  follows  that 


-      log  ©!(«)=      -  *»«i*(«  +  K)  =       -  fc2 


Similar  results  may  be  derived  for  H(u)  and  HI(W). 

The  functions  @(w),  ®iCw),  etc.,  when  for  u  is  written  the  integral  of  the 
first  kind  u(z,  s),  are  functions  of  z,  s,  but  not  one-valued,  since  u(z,  s)  is 
not  one-valued  in  z,  s.  But  from  the  formulas  just  written  it  is  seen  that 
the  second  logarithmic  derivatives  of  these  functions  are  rational,  and 
consequently  one-valued  in  z  alone  (i.e.,  the  s  does  not  appear). 

This  is  fundamental  in  the  derivation  of  the  Weierstrassian  theory,  which 
we  shall  consider  in  the  next  Chapter. 

EXAMPLES 


1.   Show  that  E(k,-\=E=  j     dn2(u,  k)du, 

2 


o 
E'=   C*  dn2(u,k')du. 

2.  Through  the  definitions  of  the  zeta-functions  of  Art.  253  derive  independently 
the  formulas  given  in  Chapter  X  for  &i(it)j  Hj(w)  and  H(tt). 

3.  Prove  that  iZw(iu,  k)  =  Z00O,  fc')  + 
and                                     iZM  (iu,  k}  =  Z1n(tt,  k')  + 


2  TT 

4.   Prove  that     Z^  =  — 
K 


2  TT 

K  l  +  2?cos  cosgcos.  -  • 

K  K  K 

Derive  similar  expressions  for  Z10(w)  and  Zu(w). 

(Thomae,  Sammlung,  etc.,  p.  16.) 


(        \m    •     m7tu 

oo  (     q)     sin  — 

r     i-g2w 

/V                                            j?V 

I  sin    ^ 

ELLIPTIC    INTEGRALS   OF   THE    SECOND   KIND.          297 

5.   Verify  the  results  indicated  in  the  table : 


iK' 


zw  +  ^ 

0 

oc 

?  77 

X 

0 

Z10  +  .^ 

0 

0 

Z»  +  ^ 

0 

0 

Zoo 

0 

0 

Z01 

0 

0 

Z10 

0 

OC 

Z« 

CiC 

0 

6.   Show  that 


*<*""* 


V7!  - 


7.    Prove  that 


Roberts  (Liouvitte's  Journ.  (1),  Vol.  19), 
Wangerin  (ScMomtich's  Zeit.,  Bd.  34,  p.  119). 


8.    Complete   the   table   of    Ex.  5  by  letting  u  take  values  i  K,  K  +  J  i'A', 
4  iA',  ^  A  +  i  iAr,  |  A  +  |  iK',  etc. 


CHAPTER   XIV 
INTRODUCTION  TO  WEIERSTRASS'S  THEORY 

ARTICLE  255.  In  the  previous  study  we  have  followed  the  historical 
order  of  the  development  of  the  elliptic  functions  and  have  made  funda 
mental  Legendre's  normal  form.  We  may  just  as  well  use  the  one  adopted 
by  Weierstrass, 


V4(t-  ei)(t-  e2)(t-  e3) 

where  V±(t-ei)(t-  e2)(t-  c3)  =  V/S(0     (see  Chapter  VIII) . 

We  have  taken  infinity  as  the  lower  limit,  because  this  value  of  t,  as  we 
shall  later  see,  corresponds  to  the  value  u  =  0.  We  saw,  Art.  185,  that 
this  integral  could  be  transformed  by  a  simple  substitution  into  the  normal 
form  of  Legendre.  Consequently  in  the  derivation  of  the  new  formulas 
we  need  not  always  return  to  the  consideration  of  the  Riemann  surface, 
but  in  this  respect  we  may  rely  upon  our  former  developments. 

ART.  256.     If  in  the  above  integral  we  write  (see  Art.  195) 

it  follows  immediately  that 


du 

In  Art.  185  we  saw  that  the  transformation  of  Weierstrass 's  normal  form 
to  that  of  Legendre  is  effected  through  the  substitution 

t  =  e3  H >     where     e  = 


el  -  e3 

We  therefore  have  1 

<$u  =  e3  H  --  -  -- 


Since  sn  I  -^=  )  is  a  one-valued  function,  the  function  <@u  must  also  be  one- 
Vv«/ 


_ 

valued;  and  since  sn2(u  \/ei—  e3)  has  the  periods  2  \/eK  and  2 
these  are  also  the  periods  of  <@u. 
We  put  (Art.  196) 

2  VI  K  =  2a),     2  V~eiK'  =  2  co', 

so  that  the  function  $>u  has  the  periods  2  co  and  2  a>'.     We  further  note 
that  sn2u  being  an  even  function,  the  same  is  true  also  of  <@u. 

298 


INTRODUCTION   TO    WEIERSTRASS'S   THEORY.  299 

ART.  257.  As  we  have  introduced  the  new  function  $>u  in  the  place  of 
sn  u,  following  Weierstrass  we  shall  introduce  new  functions  for  the  0- 
functions,  which  new  functions  are,  however,  closely  connected  with  the 
©-functions. 

If  in  the  formula  of  Art.  254 


we  put  u  +  iKf  in  the  place  of  u,  we  have 


=    ~     log  H(tt)- 


sn2u      K 
Since  p(v\/e)=  e3H  --  —  > 


£snv 


it  follows  that  f(v  VI)  =  ea  +  -i  -  -  d* 


K      e         dv2 
Noting  the  identity 


or 


it  is  clear  that 


Writing  v  \/£  =  u,  this  formula  becomes 


which  is  a  one-  valued  function  of  z  (see  Art.  254). 
We  thus  have 


or,  if  we  put 

ou  =  fie   2  ^      £         ] 

where  /9  is  a  constant,  then  is 

-  &u  =  — -  log  ou. 
du2 

The  arbitrary  constant  /?  we  may  so  choose  that  in  the  development 
of  ou,  the  coefficient  of  the  first  power  of  u  is  unity. 


300  THEORY   OF   ELLIPTIC    FUNCTIONS. 

By  Maclaurin's  Theorem  this  development  is 

ou  -  <r(0)  +  ua'(Q)  +  •    •    -  . 
Since  H(0)=  0,  we  also  have  <r(0)=  0;  and  noting  that 


A/e 

we  have  o 

</(())  =  1  =  JL-H'(0). 

It  is  thus  shown  that  ,  /- 


H'(0) 
and  consequently  /?     _  1  /    .  I  - 


If  we  differentiate  the  expression 

1 


wehave  1 

cnudnu  =  —^ 

Vk 
Writing  u  =  0,  it  is  seen  that 


or 


\//c      02(0)  Hr(0) 

It  follows  from  above  that 


ART.  258.     The  expression 

d2  log  av 

becomes  when  integrated 


where  the  lower  limit  w  and  the  constant  y  are  connected  as  follows : 
If  we  define  the  small  zeta-function  by 

rv^—     (see  Art.  277), 
av 

we  may  write  jv 

av 


INTRODUCTION   TO   WEIERSTRASS'S   THEORY.  301 

Putting  v  =  a)  in  this  formula,  we  have  at  once 

'cu  =  —  (at)  =  rj  =  -   I    $(v)dv  +  TJ. 
o  J  \> 

We  may  similarly  introduce  the  new  quantities 


If  we  put  (see  Arts.  195  and  256) 

pv 
it  follows  that 


=  t,   dv  =  —=,  &w  =  ei,     #w"  =  e2,     &*>'=  e3, 


VS(t) 

and 


or 

tdt 


In  a  similar  manner  as  in  Art.  194,  it  is  seen  that 

9    I  —    R       along  the  upper  bank  of  the  canal  c^e? 

J    V  in  the  upper  leaf;  and 


tdt 

-  =  A        (upper  leaf)  , 
VS(t) 

where  B  denotes  the  difference  in  the  values  of  the  integral 


on  the  right  and  the  left  bank  of  the  canal  b,  and  A  the  corresponding 
difference  on  the  left  and  the  right  bank  of  the  canal  a. 
If  any  arbitrary  path  of  integration  is  taken,  we  have  * 


rtdt         B 
.  V^m     2 


=      +  m'A  +  I'B, 
*  VS(t)       2 

where  m,  /,  m' ',  /;  are  integers. 

*  See  Bruns,  t/e&er  die  Perioden  der  elliptischen  Integrals  erster  und  zweiter  Gattung, 
Math.  Ann.,  Bd.  27,  p.  234. 


302  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  follows  from  above  that 

and 


the  congruences  being  taken  with  regard  to  integral  multiples  of  A  and  B. 
ART.  259.     By  definition  of  Art.  257  we  have 


It  follows  that 


From  the  formulas  H(w  +  K)  =  HI(W) 

and  H'(tt  +  X)=Hi'(M) 
we  have  at  once  H(K)  =  Hi(0) 

and  H'(K)=H/(0)=0. 

It  is  seen  that 


Further,  since  J  =  K  -  E     and     o>  = 

we  may  write  * 


Further,  since  \/eiKf=  to', 

,      ofajf  I         1  J\    ,  ,      1 

--  - 


,      ofajf  I         1  ,  , 

we  have  if  =  -  =  -  (  e3  H  ---  w  H  --  -= 

aw7  V         t  K/         V^ 

or,  since  (see  Art.  247) 


we 

(2) 
or, 
(20 


*  See  Schwarz,  Zoc.  cif.,  p.  34. 


INTRODUCTION   TO    WEIERSTRASS'S   THEORY.  303 

It  follows  at  once  from  (2')  and  (!')  that 


From  the  formulas 

we  have 

1     W(K  +  iK') 

x/7  H(K  4-  iK') 
Further,  since  (Art.  247) 


it  follows  that 
(3) 


From  the  formulas  (1),  (2)  and  (3)  it  is  evident  that 

f  +  t'-  *"• 

It  is  seen  from  the  preceding  article  that 


and  since  —      -r 

,      Jo        A 

l-^-J.—  2 

we  further  have  — 

'S2 

and 


the  congruences  being  taken  with  respect  to  the  moduli  of  periodicity  of 
the  integral  of  the  second  kind. 

We   also   have   the   relation   corresponding   to    Legendre's  formula   of 
Art.  249, 

yo)'-  TI'UJ  =  ~ 

We  may  note  that 

£(u  +  2  co)  =  £u  +  2  i), 

£(u  +  2o>')=  C"  +  2)?/; 

for  pu  being  an  even  function,  its  integral  £u  is  odd,  and  writing  u  =  —  aj 
and  —  a>f  respectively  in  the  two  formulas  just  written,  we  establish  their 
existence. 


304  THEORY    OF   ELLIPTIC   FUNCTIONS. 

ART.  260.     We  have  already  derived  the  formulas 


and  T     ,   1  J 

If  we  put 

then  is  ou 

from  which  it  is  seen  that  ou  is  an  odd  function,  the  function  H  being  odd. 
It  follows  immediately  that 

o(u  +  a>)  =  [3e2riw(v+®2  H(2  Kv  +  K) 
The  following  new  notation  is  suggested: 


where  /?i,  /?2  and  /?3  are  constants.* 

It  is  seen  that  o\u,  a2u  and  o3u  are  even  functions.     We  shall  so  deter 
mine  /?i  that  for  2  cov  =  u  =  0  we  have  <7i(0)  =  1.     We  thus  have 

i-AHiW,     or     0! 

and  similarly 

-      and 


ART.  261.     It  is  evident  from  the  previous  Article  that 
a(u  +  w) 


where  Ci  is  a  constant.     For  v  =  0,  it  is  seen  that  C\=  oat,  and  conse 
quently 


0(J) 

We  further  have 

2     (     -  - 
~ 


o(u  +  at')  =  (3e  2       H(2  K 


or 


These  constants  are  expressed  through  Weierstrassian  transcendents  in  Art.  345. 


INTRODUCTION   TO   WEIERSTRASS'S   THEORY.  305 

Writing  u  =  0  =  v  and  noting  that  2  r)a>'  —  -i  =  2  r)'a>,  it  is  seen  that 


It  also  follows  without  difficulty  that 


00) 


The  functions  oiu,  0*2,11,  o%u  are  like  ou,  one-  valued  functions  of  u,  that 
have  everywhere  in  the  finite  portion  of  the  plane  the  character  of  integral 
functions. 

ART.  262.     From  the  formulas  above  we  have 


or 


0(2K 

^cn'22Kv. 


and  similarly 


fef)2=a 

(02U\2  =  b,dn22KVf 


\auj 

where  a,  a',  bf,  c'  are  constants. 

Since 

9U  —  e'. 

it  is  evident  that 


where  c3  is  a  constant. 
If  we  put 


we  have 
so  that 


—  63 


sn22Kv 


e1- 


or 


<tu 


c\(®u  —  e\),   where  GI  is  a  constant. 


In  the  same  manner  we  have 


-  k2  = 


dn22Kv 


\au/ 


_  62^   where  c2  is  a  constant. 


306  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

We  have  accordingly 

v  @u  —  e\  =  cti 

au 

au 

au 
where  d\,  d2,  and  d3  are  constants. 

To  determine  the  constants  we  note  that  au  may  be  developed  in  the 
form 

au  =  u  +  b3u3  +  b5u5+   •  •   •  , 
and  also  that 

oku  =1  +  b2,ku2+  64,*  u*+    ...     (k  =  I,  2,  3), 

where  the  b's  are  definite  constants. 
We  therefore  have 

OkU 1      1  -f-  b2tku2-\-  •  •   - 1 

au       u      1  +  b-)U2+   ...       u 


In  the  neighborhood  of  the  point  u  =  0  we  also  have 
sn  v  =  v  4-  esv3  +  «   •  •  , 


sn^==^= 


so  that 


and 


Since 


it  follows  that 

1 


I 

$>u  —  e\  =  — -  +  HQ  -\-  63  — 
uz 

On  the  other  hand  we  had 


+dltku  +  .  -  .         (jfc  =  1,2,3). 
u  \ 


It  follows  that  rffc2=  1  or  dfc  =  ±  1,  and  consequently 


INTRODUCTION   TO   WEIERSTRASS'S  THEORY.  307 


Since  the  quotient  ^^  is  a  one-valued  function,  we  may  take  the  positive 

au 
sign  (see  Schwarz,  loc.  cit.,  p.  21). 

We  further  have  _ 

Cu  \      \/e\  —  e3         1     au 
—  -]=  —  =====  —  -  =  -- 
V«/       Vf?tt  -  63       V  £  ^3^ 

Similarly  it  is  seen  that 


or 

and  also  that 


ART.  263.     It  follows  from  the  formulas 


that 

^2(^7^3"  •  w)  =  1 
Further,  since 


_,. 

azu  =  e  -i  u  —  -  -  —  "  » 

Oil) 

we  have 

72  J2 

-—  log  a3u  =  -—  log  a(u  +  a}')  =  -  %>(u  +  &')  =  -  p(u  -  a)'). 
du2  du2 

Admitting  the  relation  (see  Art.  316) 


we  have 


d/asu  +        \ 
au\  cr3u  / 

Since 

it  is  seen  that  * 


E(u)=  C 
Jo 

u)  =  -±=  l^L  +  eiu\ 

Vel-e3\(>3u  I 

*  See  Schwarz,  loc.  cit.,  p.  52. 


308  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Further,  since  (Art.  259) 


E  =      ZlZ.    and    K  =      ei-  e3  •  w, 

V6!-  63 

it  follows  from 

Z(u)  =  E(tO-uf 
K 

that 


V  ei  -  e3  V  ff3tt  /      V  el  -  e3  W 

1  /fr3'^  _   7£W\ 

Vei-  e3  \  <*3^        w/ 
The  last  formula  may  be  written  * 

+aj')-riu-  /I. 
u  J 


-  e3 

EXAMPLES 

1.   Jacobi,  Werke,  I,  p.  527,  wrote 

f 

^>  2 

9  =  am 


2  K 
show  that  x?(Q)—  £(x)  =  

71 

2.   Prove  that 


3.  Let 

Show  that 

Px  x        1     ,   l  2  ,  4, 

P(u}  =  --  1  --  u2  H  --  u4  + 
u2  15  189 

4.  If  F(k2)  is  the  coefficient  of  u2n~2  in  the  preceding  example,  show  that 


5.    Prove  that  the  function  P(u}  of  Example  3  satisfies  the  relation 

P'(w)2  =  4  p(tt)3  _  4  (j  _  p  +  j.4)  p(u)  -  /T  (1  +  A;2)  (1-2  fc2)  (2  -  /c2)  ; 

or         P'(u)2=4P(u)*-g2P(u)-g3. 

(Hermite,  Serret's  Calcul,  t.  II,  p.  856.) 

*  See  Enneper,  Elliptische  Functionen,  p.  221. 


CHAPTER   XV 
THE  WEIERSTRASSIAN  FUNCTIONS  $u,  %u,  an 

ARTICLE  264.  We  saw  in  Chapter  V  that  the  doubly  periodic  functions 
of  the  second  order  or  degree  are  the  simplest  doubly  periodic  functions. 
These  functions  are  either  infinite  of  the  first  order  at  two  distinct  points 
of  the  period-parallelogram,  or  they  are  infinite  of  the  second  order  at  one 
point  of  the  period-parallelogram.  The  first  case  has  been  considered 
in  Chapter  XI.  We  shall  now  consider  the  latter  case.  Among  this  group 
of  functions  we  shall  take  the  simplest,  viz.,  those  which  become  infinite 
of  the  second  order  at  the  origin. 

Such  a  function  may  be  expressed  in  the  form 


where  6^0,  and  where  P(u)  is  a  power  series  in  integral  ascending  powers 
of  -M. 

It  is  shown  below  that  the  constant  a  =  0.     We  therefore  have 

<£(")  _  L  |  p(") 
6         u2         b 

The  constant  term  that  occurs  hi  the  power  series  P(u)  is  put  on  the  left- 
hand  side  of  the  equation,  and  the  function  which  we  thus  have  was  called 
by  Weierstrass  the  Pe-f  unction  and  denoted  by 

g>(w)  or  more  simply  gra. 
This  function  is  of  the  form 

&u  =  -5  +  *  +(("))• 
u2 

The  "  star  "  indicates  that  no  constant  term  appears  on  the  right-hand 
side  of  the  equation,  since  it  has  been  put  on  the  left-hand  side,  and  the 
symbol  ((u))  denotes  that  all  the  following  terms  are  infinitesimally  small 
when  u  is  taken  infinitesimally  small  and  are  of  the  first  or  higher  orders. 
If  the  point  at  which  the  function  becomes  infinite  is  not  the  origin  but 
the  point  v,  we  may  transform  the  origin  to  this  point  and  consequently 
have  to  write  everywhere  u  in  the  place  of  u  —  v. 

309 


310  THEOKY    OF   ELLIPTIC   FUNCTIONS. 

We  may  show  as  follows  that  the  constant  a  is  zero:  We  had 

4>M  =  4  +  -  +  c  +  ciu  +  c2u2+  c3u3  +  .... 
u2      u 

Consider  also  the  function  <j>(—  u).  It  is  doubly  periodic,  having  the  same 
pair  of  primitive  periods  as  has  (f)(u),  and  consequently  like  (j>(u)  is  infinite 
of  the  second  order  on  all  points  congruent  to  the  origin.  It  may  be 

written  h 

$(-  u)  =  JL_  «  +  c  _  ClU  +  c2u2-  .... 

u2      u 
We  therefore  have 


It  follows  also  that  (j>(u)—  <j>(—  u)  is  a  doubly  periodic  function  with 
the  same  pair  of  primitive  periods  as  <j>(u),  and  consequently  can  become 
infinite  only  where  (f>(u)  and  $(—  u)  become  infinite  and  therefore  only  on 
the  points  congruent  to  the  origin.  But,  as  seen  from  the  last  equation, 
<j)(u)  —  </>(—  u)  becomes  infinite  at  the  origin  only  of  the  first  order.  We 
thus  have  a  doubly  periodic  function  which  becomes  infinite  at  only  one 
point  within  the  period-parallelogram  and  at  this  point  of  the  first  order. 
We  have  seen  in  Art.  101  that  there  does  not  exist  such  a  function.  It 
follows  that  a  =  0;  and  we  further  conclude  that 

</>(u)—  </>(—  u)=  Constant, 

otherwise  we  would  have  a  doubly  periodic  function  which  is  an  inte 
gral  transcendent  contrary  to  Art.  83.  As  there  appeared  no  constant 
term  on  the  right-hand  side  in  the  development  in  series  of  the  function 
4>(u)—  <£(—  u),  we  conclude  that 

<£(u)-<£(-W)=0, 

or  <j>(u)=  </>(—  u). 

It  is  thus  seen  that  the  elliptic  function  of  the  second  degree  which 
becomes  infinite  of  the  second  order  at  only  one  point  of  the  period- 
parallelogram  must  be  an  even  function. 
It  follows  that 


or  (W)_C=       +M+ 

b  u2       b  b 


This  function  we  denote  by  ®u  and  we  require  that  pu  be  a  one-valued 
doubly  periodic  function  of  the  unrestricted  variable  u  which  has  the  char 
acter  of  an  integral  rational  function  at  all  points  that  are  not  congruent  to 
the  origin.  At  the  origin  and  the  congruent  points  $w  must  be  infinite  of 
the  second  order  and  is  to  be  an  even  function. 


THE   WEIEKSTRASSIAN   FUNCTIONS    ffu,  &,  <ru- 


311 


ART.  265.     We  may  next  show  that  in  reality  there  exists  a  function 
which  has  the  properties  required  of  <@u. 

Let  w  =  2fiw  -f  2(«V, 

where  //  =  0,  ±  1,  ±  2,  .  .  .   ;   a'  =  0,  ±  1,  ±  2,  .  .  .   ;  w  =  0  excluded. 
Form  the  function 

1     ,  ^         1 


This  function  does  not  have  the  properties  desired  of  pu,  since  the  series 

V is  not  convergent.     For  if  we  give  to  u  the  value  zero,  we  have 

T  (u  ~  ™)2 

V  — ,  which  is  not  convergent  (see  next  Article). 
^fw2 


But  if  we  form  the  series 


(u 


J_        .J_J 

-  w)2       w2  J 


and  impose  the  condition  that  the  minuend  and  the  subtrahend  which 
appear  in  the  difference  under  the  summation  sign  cannot  be  separated, 
then  this  series  is  absolutely  convergent  (Art.  266). 

If  we  put  an  accent  on  the  summation  sign  to  indicate  that  the  value 
w  =  0  is  excluded  from  the  summation,  we  may  write 

^  =  L+vf\—L-      ±1. 

u2     **  l(u-w)2      w2) 
ART.  266.     We  must  show  that  the  series 


6u>' 


(u-  w)2     w2 

is     absolutely     conver 
gent. 

Let  the  shortest  dis 
tance  from  the  origin  to 
any  point  on  the  periph 
ery  of  the  parallelogram 
passing  through  the 
points  2  a>,  —  2  co,  2  w', 
—  2  a/  be  d\,  and  let  d2 
be  the  longest  distance 
from  the  origin  to  any 
point  on  the  periphery. 


Fig.  70. 


On  the  periphery  of  this  parallelogram  there  He  8  =  32  -  I2  period- 
points.     For  these  points  we  have 


312 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


On  the  second  parallelogram,  passing  through  the  points  4  co,  —  4  a), 
4  w'f  _  4  &'  there  are  52  —  32  =  8  •  2  period-points,  and  for  these  we  have 

2di  ~  \  w\  ^  2d2. 

On  the  third  parallelogram  indicated  in  the  figure  there  are  72  -  52  =  8  •  3 
period-points,  and  for  them  there  exists  the  inequality  3  di  ^  \w\  ^  3 d2j 
and  for  the  n  +  1st  parallelogram  there  are  (2  n  +  3)2  —  (2  n  +  I)2  = 
8(n  +  1)  period-points,  and  for  them  we  have 

(n  +  l)di  ~  |  w  |  ^  (n  +  I)d2. 


In  the  first  parallelogram  we  have 
in  the  second  parallelogram  we  have 
in  the  third  parallelogram  we  have 


J_ 

w2 

1 


It  follows  that 


J_ 

w2 


8*1 


(2 


for  the  first  parallelogram, 

J--  for  the  second  parallelogram, 


8  •  3 


• 
=  —  —  —  for  the  third  parallelogram, 


and  consequently 


1 
2  )  12 


32 


The  series  on  the  right  is  the  well-known  divergent  harmonic  series. 
We  have  further 

01 

=  — '—  for  the  first  parallelogram, 

8-2 


w 


8-3 
(3di): 


for  the  second  parallelogram, 


for  the  third  parallelogram, 


and  consequently 


3<  jMJL  ,1  ,  _3_ 

di3n3      23      33 


which  is  absolutely  convergent.* 

*  Eisenstein,  Genaue  Untersuchung,  etc.,  Crelle,  Bd.  35,  p.  156;  Vivanti-Gutzmer, 
Eindeutige  Analytische  Functionen,  pp.  168  et  seq.;  Osgood,  Lehrbuch  der  Funktionen- 
theorie,  p.  444. 


THE   WEIERSTRASSIAN   FUNCTIONS    pu,  &>,  all.  313 

ART.  267.     We  may  next  show  that 


(u  -  w)2 
is  absolutely  convergent. 

We  limit  u  to  the  interior  of  a  circle  with  radius  R,  where  R  is  arbi 
trarily  large,  but  finite.  With  2  #  as  a  radius  a  circle  is  described  about 
the  origin.  Within  this  circle  there  is  only  a  finite  number  of  points  w. 
Any  of  these  quantities  w  situated  within  or  on  the  circumference  of  the 
circle  with  radius  2  R  is  denoted  by  w',  so  that 

We  denote  any  of  the  points  w  without  the  circle  by  w"  so  that 

I  -../'  I  \  o  P 
I  W     I  >  Z  K. 

It  is  clear  that 

w  w'  u/ 

The  series 


(M  -  w)2       w2 

w 

is  composed  of  a  finite  number  of  terms  and  has  a  finite  value  if  u  does 
not  coincide  with  any  of  the  values  w. 

It  is  seen  that  this  series  has  the  character  of  an  integral  rational  func 
tion  and  is  continuous  for  all  points  except  u  —  w'  which  are  situated 
within  the  circle  with  radius  2  R. 

We  consider  next  the  series 


ur 

and  limit  u  to  the  interior  of  the  circle  with  radius  R  about  the  origin  as 
center. 
We  then  have  u         \ 

rf~'    <  2 
We  also  have 


1     (         1 


(v/>-u)2      rfvj,        ^y 

(\        u/')  \ 

and  since  u  I 

the  expression  may  be  developed  in  the  series 


(w"~ 
or 


314  THEORY   OF   ELLIPTIC   FUNCTIONS. 

By  reducing  all  the  terms  to  their  absolute  values  we  have 


1 


R 


1 


(w"-u)2      w"2 


The  expression  in  the  braces  converges  towards  a  definite  limit,  G,  say. 
It  follows  that 


w 


w" 


which  we  saw  above  is  an  absolutely  convergent  series.     It  follows  that 


w" 


(u  -  w"}2      w"2 


is  a  finite  quantity,  and  since 

r 


w' 


\ 


(u  -  w'Y       wft- 


is  a  finite  quantity,  it  is  seen  that 


is  absolutely  convergent  within  any  finite  interval  that  is  free  from  period- 
points.  The  series  is  also  seen  to  be  uniformly  convergent  within  (Art.  7) 
the  same  interval. 

We  have  thus  shown  that  the  function 


u 


(u  -  w) 


w 


(w  =  2uw  +  2/*V:M  =  0,  ±1.  ±2,  .  .  .  ;  w  =  0  excluded) 
\  jf*  I 

has  only  at  the  points  u  —  w  (including  w  =  0)  the  character  of  a  rational 
(fractional)  function;  at  all  other  points  it  has  the  character  of  an  integral 
(rational)  function.  At  the  points  u  =  w  the  function  becomes  infinite  of 
the  second  order. 

ART.  268.     In  order  to  show  that  the  function 

1  1  ) 


corresponds  completely  with  the  function  g?w  defined  in  Art.  264  we  must 
first  show  that  it  is  doubly  periodic. 


THE   WEIERSTRASSIAN   FUNCTIONS   W,  fu,  <m.  315 

Since  the  expression  is  uniformly  convergent,*  we  may  differentiate 
term  by  term  and  have 

fu  =  -  -  -  2  £'       l         -  2  Y_J_, 

u3      7  (u  -  w)3      yKu-  w)*' 

or 


<&'u  =  -  2  V (w  =  0  inclusive). 

±T  (u  -  w)3 


It  follows  that 


From  this  it  is  seen  that  the  totality  of  values  on  the  right-hand  side  is 
not  altered  provided  the  series  is  absolutely  convergent,  and  consequently 

jp'(M  +  2w)  =  tfu. 
In  a  similar  manner  we  have 


We  have  thus  shown  that  the  function  p'u  is  a  doubly  periodic  function 
which  is  infinite  of  the  third  order  for  u  =  0  and  for  the  congruent  points. 
For  all  other  points  this  function  has  the  character  of  an  integral  function. 
We  may  prove  that  the  series  ]jT  -  -  —  is  absolutely  convergent  as 
follows:  As  above  write 


(u  —  w)3    y  (u  —  w')3    y  (u  —  w")3 

The  series  ^ — —  has  a  finite  value  if  u  does  not  take  one  of  the 

values  w'.     To  show  that  2) 77-^  is  convergent,  we  note  that 

w"   fa  ~"    W    ) 

-  1    =  1       1 
(u-w)3     ^/^y' 

\         wl 
and  since 


we  have 


<• 


1 


(u  -  w") 


< 


8 


n"    I    3 


also,  since  5^  -  -,  as  shown  above,  is  convergent  for  all  values  of  w" 

"3 


) 

except  w"  =  0,  it  follows  that 


* 


is  absolutely  convergent. 

(u  -  w")3 

Osgood,  Lehrbuch  der  Funktionentheorie,  pp.  83,  258. 


316  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  269.     We  have  at  once  from  the  formulas  above 

$>'(u  +  2co)du  =  p'udu, 

and  consequently  also 

p(u  +  2  co)  =  <@u  +  c. 

Similarly  it  is  seen  that 

p(u  +  2  to')  =  pu  +  cf. 

Since  @u  is  an  even  function,  its  first  derivative  <@'u  is  necessarily  odd,  so  that 

If  then  we  write  —  co  for  u  in  the  formula  *  above,  we  have 
$>co  =  §?(—  co)  +  c,     so  that  c  =  0. 

Similary  it  is  seen  that  c'  =  0. 

ART.  270.     We  may  derive  as  follows  another  proof  that  <@u  is  doubly 
periodic  without  making  use  of  its  first  derivative. 

The  formula 

1    ,    v*'( 


1 \_\ 

w)2       w2  } 


becomes,  if  w  is  changed  into  —  w, 

1 


The  term  which  corresponds  to  w  =  —  2  co  is  taken  without  the  summation 
sign.     The  sum  taken  over  all  the  values  of  w  except  w  =  0  and  w  =  —  2  co 
is  denoted  by  2  *. 
We  thus  have 

1       i          1  1 


u2       (  (u  -  2co)2       (2co)2)       %    ((u  +  w)2      w2 

The  totality  of  the  values  of  w  under  the  summation  sign  is  not  changed  if 
we  write  —  w  —  2  co  instead  of  w. 
It  follows  then  that 

1  1  ) 


(u-w-2co)2       (w  +  2co)2 
Adding  these  two  expressions  and  dividing  by  2,  we  have 


i     ? 

2  co)2  ) 


2  (u  +  w}2       (u  -  w  -  2  co)2       w2       (w  + 

*  See  Osgood,  loc.  cit.,  p.  444;  Humbert,  Cours  d'  Analyse,  t.  II,  p.  194. 


THE    WEIERSTKASSIAN   FUNCTIONS   ?M,  ?M,  <ru.  317 

In  this  formula  write  —  u  for  tt;  then  since  pu  is  an  even  function,  it  is 
seen  that 

+  ; i 

u2 

i  111 


(u  -  w)2       (u  +  w 
Finally,  changing  w  into  u  —  2  w,  we  have 

(ii) 


1 


((u-  w  -2w}2       (u  +  w}2       w2       (w  +  2  to)2 
Comparing  the  formulas  (II)  and  (I),  it  follows  that 

p(u  -  2  w)  =  $u, 
or  writing  u  +  2  a>  for  u, 


In  a  similar  manner  it  may  be  shown  that 

pu  =  p(u  +  2o>')- 

ART.  271.  It  is  evident  from  the  formulas  above  that  2a>,  2  a)'  form,  a 
primitive  pair  of  periods  of  the  argument  of  the  function  $u.  The  parallel 
ogram  with  the  vertices  0,  2  at,  2  a>',  2  CD  +  2  a/  is  free  from  periods,  since 
all  the  quantities  -a?  represent  points  that  are  congruent  to  these  four  points. 
If  we  select  the  pair  of  periods  2  M,  2  a}',  we  may  bring  them  into  promi 
nence  by  writing  pu  in  the  form 


If  a  transition  is  made  to  an  equivalent  pair  of  periods,  we  write 
2  5  =  2  pw  +  2  qw',     2a)  =  2p'u)  +  2  g'o/, 

where  pqf  —  qp'  ~~  1  (p,  q}  p',  q/  being  integers). 

It  is  clear  (Art.  80)  that  the  totality  of  values  w  remains  unaltered  by 
this  transformation  and  consequently  we  have 


It  is  thus  seen  that  jpu  remains  unchanged  by  a  transition  to  an  equivalent 
pair  of  primitive  periods. 


318  THEORY   OF   ELLIPTIC    FUNCTIONS. 


THE    SlGMA-FUNCTION. 

ART.  272.     By  integrating  twice  the  gw-function  we  may  derive  another 
important  function. 
It  is  clear  that 


or  -   Cpu  du  =  -  +  V  '  \  —  —  +  ~  I  +  Constant. 

J  u      **  (u  —  w      w2) 

The  sum  of  the  terms  on  the  right-hand  side  is  not  convergent,  but  it 
may  be  made  convergent  by  a  proper  choice  of  the  arbitrary  constant. 
For  writing 


w 


we  shall  show  that  this  expression  is  absolutely  convergent  and  becomes 
infinite  of  the  first  order  only  at  the  points  u  =  0  and  u  =  w. 
It  is  seen  that 


u  —  w          w 


w 


w 


u  —  w      w      w2          w3  w      w2 


As  in  Art.  268,  it  may  be  shown  that  the  series  is  convergent,  so  that  the 
above  development  of  —  I  <@u  du  is  convergent. 

It  is  also  seen  that  the  above  series  is  infinite  only  of  the  first  degree  at 
the  origin  and  its  congruent  points.     It  follows  that  —  /  @u  du   cannot 

be  doubly  periodic. 

Integrating  again  the  above  expression  we  have 


where  we  have  introduced  the  constant  of  integration  under  the  logarithm 
which  comes  after  the  summation  sign. 

We  shall  next  show  that  this  expression  is  also  absolutely  convergent  if 
u  does  not  coincide  with  one  of  the  periods  of  pu. 

To  do  this  we  limit  u  to  the  interior  of  a  circle  with  radius  R,  where  R 
is  arbitrarily  large  but  finite. 


THE   WEIERSTRASSIAN   FUNCTIONS    &U,  &,  <ru.  319 

The  quantities  w  we  again,  Art.  267,  distribute  into  two  groups,  so  that 


We  then  have 


2R, 


1C' 


>2R, 


<l 


where  the  first   summation  on  the  right   consists  of  a  finite  number  of 
terms,  and  is  consequently  finite  so  long  as  none  of  the  logarithmic  terms 
which  appear  is  infinite,  that  is,  so  long  as  u  does  not  coincide  with  one  of 
the  quantities  w'. 
Noting  that 


1/1        u\          u        1  1  u\2      l  I  u  \3 
10g(  l-^)=-^-^l~^)~ 


it  is  seen  that 


which  is  an  absolutely  convergent  series  (Art.  268). 
It  follows  that 


—  I  du  I  pu  du 


is  absolutely  convergent  for  all  values  of  u  other  than  u  =  0  and  u  =  w. 

Since  the  logarithmic  function  is  many-valued,  the  above  integral  func 
tion  is  many-valued.  To  avoid  this  difficulty  we  no  longer  consider  this 
function  but  the  one-valued  function 


-fdufpudu 


au  =  e 


This  sigma-f unction  is  therefore  expressed  as  a  product  of  an  infinite 
number  of  factors.  As  shown  in  a  following  Article  this  product  is  abso 
lutely  convergent  if  the  two  factors  that  occur  under  the  product  sign 
are  not  separated.  The  agreement  of  this  function  with  the  function 
defined  in  Art.  257  follows  in  the  sequel. 

The  function  au  is  one-valued  and  becomes  zero  at  the  origin  and  at 
the  points  congruent  to  the  origin.  The  accent  on  the  product  sign 
denotes  that  the  factor  which  corresponds  to  w  =  0  is  excluded.  The 
sign  o  is  chosen  on  account  of  the  similarity  of  this  function  with  the 
sine-function. 


320  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  is  seen  at  once  that 


The  function  ou  is  not  doubly  periodic.  It  has  like  the  theta-f  unctions 
for  all  finite  values  of  u  the  character  of  an  integral  function  and  may  be 
expressed  as  an  absolutely  convergent  power-series  with  integral  positive 
exponents  (Arts.  276,  336).  Like  the  function  pu  it  is  not  changed  when 
a  transition  is  made  from  one  pair  of  primitive  periods  of  the  function 
<@u  to  an  equivalent  pair. 

ART.  273.     Historical.  —  Eisenstein  (in  Crelle's  Journal,  Bd.  27,  p.  285, 
1844)  formed  the  product 


where  A  and  A'  are  quantities  such  that 

4;  =  a  +  ip     (/?  ^  0), 

-A 

while  n  and  /*'  take  all  values  ±  1,  ±  3,  ±  5,  •  •   •   ;  and  on  page  287  he 
formed  the  products 


(>U')=  ±2,  ±4,  ±6,  •   •  .  , 
P  =  ±  1,  ±  3,  ±  5,  -   -  -  . 

On  page  288  Eisenstein  says  that  the  quotient  of  any  two  such  products 
gives  rise  to  the  doubly  periodic  functions  and  he  closes  the  article  with 
the  remark: 

"  Die  hier  angestellte  Untersuchung  ist  ubrigens  so  elementar  Natur,  dass 
sie  sich  wohl  eignen  mochte,  den  Anfanger  in  die  Theorie  der  elliptischen 
Functionen  einzufuhren  .  '  ' 

In  Crelle's  Journal,  Bd.  30,  p.  184,  Jacobi  called  attention  to  the  fact 
that  Eisenstein  had  formed  defective  ©-functions  owing  to  the  fact  that 
the  above  products  are  not  absolutely  convergent.  Jacobi  at  the  end  of 
this  article  claims  that  the  "exact  formulas"  are  given  (by  Jacobi)  in 
Crelle's  Journal,  Bd.  4,  p.  382;  Werke,  Bd.  I,  p.  297  (see  also  Werke,  Bd.  I, 
p.  372). 

Cayley  (Elliptic  Functions,  p.  101)  remarks  that  such  products  as  the 
above  "in  the  absence  of  further  definition  as  to  the  limits  are  wholly 
meaningless;  "  but  Cayley,  loc.  cit.,  pp.  301-303,  fixed  these  limits  (see  also 
Cayley,  Camb.  and  Dublin  Math.  Journ.,  Vol.  IV  (1845),  pp.  257-277,  and 
Liouville's  Journal,  t.  X  (1845),  pp.  385-420),  and  illustrated  them  by 
means  of  a  "bounding  curve." 


THE   WEIERSTBASSIAN   FUNCTIONS    VU,  Jfc,  ru.  321 

It  may  be  observed  that  the  above  remarks  are  applicable  also  to  the 
infinite  products  of  Abel  (Recherches  sur  les  fonctwns  elliptiques,  Crelle, 
Bd.  2,  p  154;  (Euvres,  t.  I,  p.  226)  and  of  Jacobi,  Fund,  nova,  §  35;  Werke, 
I,  p.  141. 

Professor  Klein,  Theorw  der  elliptischen  Modulfunctionen,  Bd.  I,  p.  150, 
calls  attention  to  the  fact  that  the  quantities  pu,  p'u,  g2,  #3,  e\,  e2,  e$  are 
defined  by  Eisenstein,  Genaue  Untersuchung  der  unendlichen  Doppel- 
produkte,  aus  welchen  die  elliptischen  Funktionen  als  Quotienten  zusammen 
gesetzt  sind,  Crelle,  Bd.  35  (1847),  pp.  153-274,  and  Mathematische  Ab- 
handlungen,  pp.  213-334. 

We  also  note  that  the  relation 


is  the  identical  relation  given  by  Eisenstein,  Crelle,  35,  p.  225,  formula  (5). 
On  page  226,  Eisenstein  derives  the  normal  integrals  of  the  first  and 
second  kinds  in  the  forms 


f  d"  and  -   C 

J  2  V(u-  a\(n  -  a'Mu  -  a"}  J  ' 


2  V(ij-  a)(y  -  a')0/  -  a")  ^  2  V(y  -  a)(y  -  a')(y  -  a"} 

It  also  appears  from  this  paper  that  Eisenstein  had  some  idea  of  the  nature 
of  the  quantities  g2  and  #3  whose  invariantive  properties  were  discovered 
by  Cayley  and  Boole  in  1845. 

Weierstrass,  recognizing  the  true  nature  of  these  invariants,  was  the  first 
(cf.  Klein,  loc.  cit.,  p.  24)  to  make  the  Theory  of  Elliptic  Functions  from 
the  standpoint  of  the  infinite  products  and  series  as  given  in  this  Chapter 
(and  developed  by  him)  of  consequence,  and  so  he  is  to  be  considered  the 
founder  of  this  theory. 

In  his  last  lectures  Professor  Kronecker,  Theorie  der  elliptischen  Func- 
tionen  zweier  Paare  reeller  Arguinente  (W.  S.,  1891),  especially  empha 
sized  the  Eisenstein  theory  and  made  paramount  a  certain  function  En 
(denoting  Eisenstein's  name)  which  is  a  generalized  ^-function. 

ART.  274.     \Ye  saw  in  Chapter  I  that  the  infinite  product 


(1  +  av)  is  absolutely  convergent  if 

v=   1  v=l 


a, 


is  absolutely  convergent. 

To  prove  the  absolute  convergence  of  the  infinite  product  through 
which  the  sigma-f unction  is  expressed  let  \u  <  R,  \  w'  \  =  2  R, 
\w"  \  >  2  R  as  above.  We  omit  from  the  infinite  product  all  those 
factors  which  correspond  to  the  quantities  w'.  Such  factors  being  finite 
in  number  exercise  no  influence  upon  the  question  of  convergence. 


322 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


The  factors  remaining  in  the  product  are  of  the  form 


U  I     V? 


1      U2 


Since 


or  finally 


<  1,  we  may  develop  the  logarithm  in  a  power  series  and  have 

z 

_jU__ljM2__ljW3__    ...    +^-+-    "2 


1     V3    ii       3ji_4.3_^-  + 

~  3  w773  4  w"      5  w"2 


or 

Since 


-,  this  expression  is 


<  e' 


w" 


and  consequently 


«; 


1.2 


w" 


It  is  thus  seen  that  the  quantities  in  the  sigma-function  corresponding 
to  aw  above  are  such  that 


or  finally 


It  follows  that 


' 


16       16~2 


.16 

<15 


to" 


t 

"  16 


which  we  saw  above  was  absolutely  convergent.  To  the  S  |av|  we  must 
add  the  quantities  |ov|  which  correspond  to  the  quantities  «/;  but  the 
convergence  is  unchanged  by  the  addition  of  these  terms.  It  follows  that 
the  product  through  which  the  sigma-function  is  expressed  is  absolutely  con 
vergent.  Since  an  absolutely  convergent  infinite  product  is  only  ^zero 
when  at  least  one  of  its  factors  becomes  zero,  it  is  seen  that  au  vanishes 
only  at  the  points  u  =  0  and  u  =  w  and  at  these  points  au  is  zero  of  the 
first  order. 


THE  WEIERSTKASSIAN   FUNCTIONS   &u,  &,  <ni.  323 

ART.  275.     Other  properties  of  the  sigma-f unction  may  be  developed  as 
follows: 

We  have  i 


w 


If  w  is  changed  into  —  w  the  product  is  not  altered,  and  we  have 


It  follows  that 

a(—  u)  =  —  au, 

and  consequently  the  function  au  is  an  odd  function. 

ART.  276.     We  shall  consider  next  more  closely  the  form  of  the  develop 
ment  of  au.     In  the  product 

U         III* 


we  join  any  two  factors  that  correspond  to  opposite  values  of  w  and  thus 
have  * 


the  star  denoting  that  of  every  pair  of  values  w  and  —  w  only  one  value 
is  to  be  taken. 
It  follows  that 


If  u  is  chosen  smaller  than  any  of  the  values  w,  we  may  write 

1    U^        1  U^ 


and  consequently 


or 


*  Cf.  Daniels,  Amer.  Journ.  Math.,  Vol.  6,  p.  178. 


324  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  may  write  02    Q     c  V   * 

•    •  5  2,      -  22, 


where,  as  will  be  evident  from  the  sequel,  the  quantities  g2,  #3  are  the 
invariants  introduced  in  Art.  184.     It  is  also  evident  that  g2  and  g3  remain 
unaltered  when  we  pass  from  one  pair  of  equivalent  primitive  periods  to 
another  pair. 
It  is  seen  that 


the  star  indicating  that  the  term  with  u3  is  wanting.  The  function  ou 
is  an  integral  function  that  is  regular  in  the  whole  plane  and  may  be 
expressed  through  a  series  that  is  everywhere  convergent  (Art.  13). 

THE  ^-FUNCTION. 
ART.  277.     From  the  formula  just  written  it  follows  that 

log  an  =  log  j  u  -  ^75  "'"  2»  .  3^  5  .  7  "'  "  '  '  '  j 


It  is  evident  from  the  consideration  of  the  product    through  which  ou 
is  defined  that  this  series  is  convergent  within  a  circle  with  the  origin  as 
center  and  a  radius  that  passes  through  the  nearest  period-point. 
If  this  expression  is  differentiated  with  respect  to  u,  it  follows  that 

^  =  I  +  *  _  ,  _  22—^3  __  23  _  U5_  .... 
au       u  22  •  3  •  5  22  •  5  •  7 

The  quotient  ^^  is  often  denoted  by  £u  (Art.  258,  see  Halphen,  Fond. 
au 

Elliptiques,  t.  I,  Chap.  V). 

Differentiating  this  expression  again  and  multiplying  by  —  1,  we  have 


„.-*+.  +  *. 

The  series  through  which  pu,  <@'u  and  £u  are  expressed  are  convergent 
within  a  circle  which  has  the  origin  as  center  and  which  does  not.  contain 
any  period-point. 


THE   WEIERSTRASSIAN   FUNCTIONS    &U,  fy,  <ru.  325 

The  functions  %>u  and  p'u  are,  as  we  have  already  seen,  doubly  periodic, 
yu  being  an  even  and  <#u'  an  odd  function.  The  function  (p'u)2  is  an  even 
doubly  periodic  function  of  the  sixth  degree  and  is  infinite  of  the  sixth 
order  at  the  origin  and  all  congruent  points. 

ART.  278.  We  may  next  prove  that  %>u  satisfies  the  differential  equa 
tion  of  the  first  order  * 

(g/w)2  =  4(&m)3-  g2  $>u  -  gs- 
We  have 


5"u'      7'3  +  (("2)) 


and 

It  follows  that 


and  also  that 

-  g3 


We  note  that  the  left-hand  side  of  this  expression  is  doubly  periodic,  while 
the  right-hand  side  has  everywhere  the  character  of  an  integral  function. 
By  the  theorem  of  Art.  83,  such  a  doubly  periodic  function  must  be  a 
constant,  and  as  there  is  present  no  constant  term,  the  right-hand  side 
is  zero.  We  therefore  have  as  our  eliminant  equation 

(p'u)2  =  4  yPu  -  g2  pu  -  g3. 

ART.  279.     If  in  the  above  equation  we  use  Weierstrass's  notation  and 

put  pu  =  s,  and  $>'u  =  —  ^-,  Art.  256,  we  have 

du 


or 

u=  ±  I 

v  4  s3  —  g2s  -  g3 

agreeing  with  the  results  of  Chapters  VIII  and  XIV.  No  confusion  can 
arise  from  the  fact  that  here  we  have  written  s  for  the  variable  t  before 
used.  The  double  sign  is  accounted  for  by  means  of  the  Riemann  surface 
of  Art.  143. 

Since  s  =  oc  for  u  —  0,  we  may  write  this  integral  in  the  form 


JBL..JSL 

4s2      4S3 

*  See  for  example,  Humbert,  loc.  cit.,  p.  204. 


326  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  we  consider  values  of  s  lying  in  the  neighborhood  of  infinity  so  that 
^  we  mav  expan(i  the  integrand  in  a  power  series  and 


4s3 
then  integrate  term  by  term.     We  thus  have 


or  u  =  — 

v 
It  follows  that 

All  the  coefficients  of  this  power  series  are  clearly  functions  of  g2  and  g3 
with  rational  numerical  coefficients. 

When  this  series  is  reverted,  it  is  seen  that  —  may  in  the  neighborhood 

of  the  origin  be  expanded  in  powers  of  u\  and  it  is  also  evident  that 
s  =  <@u  may  be  expanded  in  the  neighborhood  of  the  origin  in  a  power- 
series  whose  coefficients  are  integral  functions  of  g2  and  g3  with  rational 

numerical  coefficients.      The  functions  ra  =     ,    '  and    loge  ou  have  the 

o(u) 

same  properties,  and  by  passing  from  the  logarithm  to  the  exponential 
function,  it  is  found  that  the  same  is  also  true  of  the  function  ou,  so  that 
the  development  of  ou  in  the  neighborhood  of  the  origin  is  such  that  all 
the  coefficients  are  integral  functions  of  g2  and  g%  with  rational  numerical 
coefficients.  The  sigma-function  is  therefore  a  function  of  u,  g2,  g%.  A 
method  of  determining  the  coefficients  of  ou  by  means  of  a  partial  differ 
ential  equation  is  found  in  Art.  336. 

ART.  280.     It  follows  from  the  equation  above  that 


or  ,. 

a2  r      20  «2      28 

Hence  as  an  approximation  (up  to  terms  of  the  order  ue)  we  have 


If  then  on  the  right-hand  side  of  the  last  equation  we  write  —  for  s,  we 

have 


Writing      <@u  =  -    +  *  +  c2u2  +  c3u*  +  c4u6  +  •   •   •  +  ctu2*~2+  -   •   -  ,  it 
follows  that  c2=A02     and     c3=  &g3. 

We  shall  express  the  other  constants  c4,  c5,  .  .  .  through  these  two  quan 
tities. 


THE   WEIERSTRASSIAN   FUNCTIONS   &u,  &,  *u.  327 

From  the  relation 

(#>'u)2  =  4  $*u  -  g2&u  -  g3 

we  have  through  differentiation 

2  tfu  tf'u  =12  p-u  <$'u  —  g2&'u, 
or,  if  we  give  to  u  such  values  that  $'u  ^  0, 

9»u  =  6  $2u  -  &  •        (Eisenstein,  Crelle,  Bd.  35,  p.  195.) 

Multiplying  through  by  u4  we  have 

(A)  u*v"u  =  6  u*j?u  -  J  sr2M4. 

From  the  equation 


-.  4- 
ii 

it  follows  that 
&'u  =  -  2-  +  *  +  2c2u  +  4c3u3+  •  •  •  +(2  J  -  2 


or 


We  also  have 

W2§?U  =  1   +  * 

or        u2$w  —  .-.. 
and  consequently 


v= 

where  we  have  written  down  only  the  terms  that  contain  u2*. 

Writing   these    values  in  the  equation    (A)   above    and    equating   the 
coefficients  of  u2*,  we  have  * 


v=2 

v  =  A-2 


(2  J  +  1XJ  -  3) 


This  is  a  recursion  formula  by  means  of  which  each  of  the  coefficients 
ca  in  the  development  of  %>u  may  be  expressed  through  coefficients  with 
smaller  indices. 

*  Cf.  Schwarz,  Formeln  und  Lehrsdtze,  etc.,  p.  11;  the  Berlin  lectures  of  Prof. 
Schwarz  have  been  freely  used  in  the  preparation  of  this  Chapter. 


328  THEORY   OF  ELLIPTIC   FUNCTIONS 

We  have,  for  example, 

or,  since  C2  =  ^V  92,  it  follows  that 

1 


24-3.52l/; 
and  similarly 

c  =        3  ff2!73 
5     24-  5-  7-11' 


Cft=    -J_/22!  +       ^     \ 
6     24-13V7        2.3-5V' 


We  may  therefore  write 


GU  =  u  +   *   _    _^|_    _    _^_  ^7- 

ART.  281.     We  saw  in  Art.  268  that 


If  we  make  the  condition  that  \u\  <  w,  we  may  write 


w1 


This  equation  differentiated  with  respect  to  u  becomes 

_L  4.  ?Jf   ,          .   .   nun~l 


(w  -  u)2       w2       u 
It  follows  at  once  that 


" 


We  note  that  all  terms  in  which  it?  appears  with  an  odd  exponent  vanish, 
since  a  value  —  w  belongs  to  every  value  4-  w. 


THE   WEIEKSTRASSIAN   FUNCTIONS    yu9  £u,  an.  329 

If  then   we   write  n  —  1  =  2^  —  2,  or  n  =  2  /  —  1,  and  compare  the 
above  expression  with 

it  is  seen  that 

It  follows  from  the  results  of  the  preceding  Article  that  Jj  — r;  may  be 


integrally  expressed   in  terms  of   <?2  and  gr3.     This  is  a  very  remarkable 
fact  (cf.  Halphen,  Fonct.  Ellip.,  t.  I,  p.  366). 
In  Art.  272  we  saw  that 

—  _  ^2  log  ou 
du2 


=  ±(-  2^\ 

du  \       ou / 


or  OILO"U  -  (o'li)2 

(ou)2 

The  function  ou  is  uniformly  convergent  for  all  values  of  u  in  the  finite 
portion  of  the  plane.  The  same  is  true  of  o'u  and  o"u.  Hence  it  is  seen 
that  %ni  may  be  expressed  as  the  quotient  of  two  power-series  that  are 
uniformly  convergent  for  all  values  of  u  in  the  finite  portion  of  the  plane. 
We  saw  in  Chapter  XI  that  the  functions  sn u,  cnu,  dnu  have  the  same 
property.  In  Arts.  262,  324-326  we  consider  the  analogues  of  these  three 
functions  in  Weierstrass's  Theory. 

ART.  282.     Another  expression  for  the  function  pu.  —  We  write  (cf .  Art.  60) 


and  we  shall  first  derive  a  function  of  t  which  behaves  at  the  origin  in  the 
same  manner  as  pu.     The  development  of  t  in  the  neighborhood  of  u  =  0  is 


co         1  '2\co  , 


or 


1>2  o>         1  -2  -3 


We  note  that  t  —  1  becomes  zero  of  the  first  order  at  the  point  u  =  0 
and  at  all  other  points  where  t  has  the  value  1.  The  totality  of  all  these 
points  is  expressed  through 

u  =  2[j.uj(u.  =  0,  ±  1,  ±  2,  •  •   •  ). 
The  function 


co         12\  co  I 
becomes  zero  of  the  second  order  at  all  the  points  u  =  2 


330  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Let  g(t)  be  an  integral  function  of  t  which  does  not  vanish  for  t  —  1. 
The  function 

(t  -  I)2 

will  therefore  be  infinite  of  the  second  order  for  the  value  u  =  0  and 
for  all  the  values  u  =  2  /*to.  Hence  this  function  behaves  at  these  points 
in  the  same  way  as  does  the  function  yu. 

We  may  write  g(t)  =  a  +  bt  +  ct2,  a,  b  and  c  being  constants.     It  follows 
that 

g(t)       =  a  +  bt  +  ct2 

(t  -    1)2  (t  -    1)2 

Uiri 

Since  t  =  e  "  ,  it  is  seen  that  t2  may  be  derived  from  t  by  writing  2  u  in 
the  place  of  u  in  the  expansion  of  t. 
Accordingly  we  have 


(t  -    1)2 

+^+     W^\2+.  .  .]+c[i+2tm  +  J_^rmV+  ,  .    1 
u)        I  »2\  co  ]  J        |_  a}          1  »2\    a)    ]  J 


_  ^27r2["-i    ,   uni   ,   1  /U7ii\2  .  ] 

~^t     "IT    auy  "        J 


We  wish  that  the  following  conditions  be  satisfied  : 
First.   The  term  which  becomes  infinite  of  the  second  order  must  be  of 

the  form—  -• 
u2 

Second.  The  term  which  becomes  infinite  of  the  first  order  must  not 
be  present. 

Third.  The  constant  term  in  the  development  of  the  function  in  powers 
of  u  must  be  zero. 

To  fulfill  the  first  condition  we  must  have 

_  ^2  a  +  5  +  c  =  j^  ^ 

2  2 


U  U 

for  the  second  condition,  we  must  put 


OJ2[,  xi  1    ,  2cxi  I      f     .  ,        N  id  1")      n 
—  —   o  --  -+.  -  ----  (a  +  6  -f  t;)  --   =  0, 
^2  [_    w  ^         a;     w  w  wj 


or  c  -  a  =  0. 


THE   WEIERSTRASSIAN   FUNCTIONS    yu,  ty,  au.  331 

From  the  first  condition  it  follows  that 


/t\ 
These  values  substituted  in  —  ^   '      cause  this  function  to  become 


(t  -   I)2  W2    (t  -   I)2 


a. 


The  constant  a  must  be  so  chosen  that  the  third  condition  above  may  be 
satisfied. 

We  note  that 


24 
and  since 


it  follows  that 


12 


and  consequently  that  the  third  condition  may  be  satisfied,  we  must  have 

Noting  that  ^  _  t-± 

2i 
it  is  seen  that  _2 


12  0,2 


=  sin 


=fl_ 


We  have  thus  shown  that  the  function 


LVr_J_  .11 

Sin2»£      3 
2w 


corresponds  in  its  initial  terms  with  the  development  of  pu,  so  that  it 
differs  from  $>u  only  in  quantities  which  become  infinitesimally  small  of 
the  first  order  when  u  becomes  indefinitely  small. 


332  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  283.     We  had 

1  ] 


The  quantities  w  may  be  distributed  into  two  groups.  The  first  group 
contains  all  values  w  for  which  //  =  0,  so  that  w  =  2  /j.  a).  The  second 
group  contains  those  w's  for  which  //  S  0,  so  that  w  =  2  p  oj  +  2  /*'o/. 
If  then  we  let  a>'  become  infinite,  the  values  w  of  the  second  group  become 
infinite,  and  we  have 


It  is  seen  from  Art.  22  that  this  expression  is  none  other  than  the  function 

(t  -  I)2' 
If  then  the  period  2  a/  becomes  infinite,  the  function  $ru  is  represented  by 

1 

"3 

AKT.  284.     We  shall  next  write  (cf.  Eisenstein,  loc.  tit.,  p.  216) 

F(t)  =  -  —  _  -  _ 

OJ2    (t  -   I)2 

and  we  shall  seek  to  express  *  %>u  through  t  even  when  the  second  period 

Ttiu 

2oj'  is  finite.     F(t)   being  a  rational  function  of    t  =  e  "•  remains    un 
changed  when  u  is  increased  by  u  +  2  co;  but  when  u  is  increased  by  2  a/ 


then  e  w  =  e  w    <.     Weierstrass  used  the  letter  h  to  denote  the  quan- 


tity  e  w  ,  which  Jacobi  denoted  by  q.     In  Art.  86  we  wrote  —  =  a  +  i/3, 

0) 

where  ?  >  0.     From  this  it  is  seen  that 


q  =      = 
and  consequently  |  ^  |  =  e-0*m 

Noting  Art.  81,  it  is  evident  that  we  may  always  choose  a  pair  of  primitive 
periods  so  that  \h\  <  1 

Since  t  becomes  h2t  when  u  is  increased  by  2  a/,  it  follows  that  when  u 
becomes  u  +  2  a/  F(Q  becomes 

becomes 
becomes 


See  also  Halphen,  Fonct.  Ellip.,  t.  I,  Chap.  XIII. 


THE   WEIEKSTKASSIAN   FUNCTIONS   <WL,  &t,  <M.  333 

If  we  consider  the  infinite  series 

(Sr)  F(t)+  F(h2t)+F(h*t)  +  ..  •  -  +  F(h2nt)  + 

then,  if  u  is  increased  by  2  a/,  each  term  becomes  the  following  term. 
Hence  the  series 

+  F(h2t)  +  F(h*t)+  •'...  +  F(h2nt)+  •  .  . 


is  a  doubly  periodic  function  having  the  two  periods  2  a),  2  CD'.  At  the 
point  u  =  0  and  all  its  congruent  points  this  function  becomes  infinite 
of  the  second  order;  for  then  t  equals  unity  or  some  even  power  of  h. 

ART.  285.  We  shall  next  show  that  this  series  is  absolutely  conver 
gent  for  all  points  except  the  origin  and  the  points  congruent  to  it. 

We  limit  u  to  a  region  in  which  \u\  <  R,  where  R  may  be  arbitrarily 
large,  but  finite.  The  quantity  t  has  everywhere  within  this  region  the 
nature  of  an  integral  function  and  is  different  from  zero. 

Further,  since 


*v  =  efev, 
it  is  seen  that 

1  1  1  =  e", 

so  that  |  1  1  becomes  a  maximum  with  //,  that  is,  with  R(—  j- 
If  we  put  u  =  to',  then  is  \  w  / 


\  w 

If  M  is  the   greatest  value  that  R()  can  take  for  values  of  u  within 


OJ 

the  region  in  question  and  m  the  smallest,  it  is  always  possible  to  find  an 
integer  no,  say,  such  that 

—  no  ,3/r  <  in 

and  M  <  n0  /?-. 

Hence  for  this  region  there  exists  the  inequality 

Q     /  n/u~i\^       o 
—  no-j"  <•»  i«j  — j <,  noJ/T, 

\    OJ    / 

and  consequently,  since  |  h   =  e~^,  it  foUows  that 


— 

Since  F(t)=  —  — •  -,  it  is  seen  that  in  the  first  term  of  the  in- 

oj-  (t  -  1)- 
finite  series  (S')  there  appears  (1  —  t)2  in  the  denominator;  in  the  second 


334  THEORY   OF   ELLIPTIC    FUNCTIONS. 

term  there  appears  (1  —  h2t)2  in  the  denominator;  in  the  third  term 
there  appears  (1  —  h4t)2  in  the  denominator;  •  •  •  . 

The  greatest  absolute  value  that  t  can  take  within  the  fixed  region  being 
<  |^~"o|,  the  greatest  absolute  value  that  h2nt  can  take  in  the  same 
region  is  <  |  h2n-^  \  .  If  then  we  choose  2  n  =  n0,  then  is  |  h2nt  \  <  1. 
In  the  series  (SO  we  separate  from  the  remaining  series  those  terms  (finite 
in  number)  in  which  h  occurs  to  a  power  less  than  UQ. 

The  denominator  in  any  of  the  remaining  terms  is 


where 

and  consequently 


We  therefore  diminish  the  denominator  of  the  terms  in  question  if  instead 
of  (1  —  h2*t)2  we  write  (1  —  |  hn°  |)2,  and  consequently  we  increase  the 
value  of  the  term  F(h2H). 

The  numerators  of  the  terms  which  have  been  separated  from  the  first 
UQ  terms  are 

h2n<>t,        h2n»+2t, 


which  is  a  geometrical  series  whose  common  ratio  is  less  than  unity.  It 
follows  that  the  series  (S')  is  absolutely  convergent  for  the  region  in 
question.  It  follows  also  (see  Osgood's  Lehrbuch  der  Funktionentheorie, 
pp.  72,  259)  that  this  series  is  uniformly  convergent  and  represents  an 
analytic  function.  The  terms 

F(t)+  F(h2t)  +  F(h*t)  + 

which  also  belong  to  the  series  (S')  but  which  were  not  taken  into  con 
sideration  above,  do  not  affect  the  question  of  convergence,  since  they 
constitute  a  finite  number  of  finite  terms. 

We  shall  next  establish  the  convergence  of  the  series 

(S")  F(t)+  F(h-2t) 

We  may  write 


-  h2t~1)2 


By  separating  a  finite  number  of  these  terms  from  the  series  (S")  it 
may  be  shown  as  above  that  the  remaining  terms  are  less  than  the  corre 
sponding  terms  of  a  decreasing  geometrical  series. 


THE   WEIERSTRASSIAN    FUNCTIONS    &u,  &,  all.  335 

It  follows  that  the  series 

+  F(h2t}- 
(S)  F(t) 

+  F(h~2t)  + 

is  absolutely  and  uniformly  convergent  in  any  interval  that  is  free  from 
the  points  u  =  0,  u  =  w. 

This  series  therefore  represents  a  one-valued  doublt/  periodic  function  of  u 
which  for  all  finite  values  of  u  has  the  character  of  an  integral  or  (fractional} 
rational  function.  At  the  points  u  =  0  o/nd  the  congruent  points  this  func 
tion  becomes  infinite  of  the  second  order. 

ART.  286.  We  note  that  F(0)=  F(*)=  0.  It  is  also  seen  that  the 
series  (S)  has  the  same  periods  and  becomes  infinite  of  the  same  order 
at  the  same  points  as  the  function  pu.  Two  doubly  periodic  functions 
which  in  the  finite  portion  of  the  plane  have  everywhere  the  character  of 
an  integral  or  (fractional)  rational  function  and  which  become  infinite 
of  the  same  order  at  the  same  points  can  differ  from  each  other  only  by  a 
constant  (Art.  83).  Hence  the  above  series  can  differ  from  <pu  only  by  a 

constant,  which  constant  it  will  appear  later  is  —  -*-• 

Further,  put  z2  for  /,  retaining  the  notation  of  Weierstrass,  as  no  confu 
sion  can  arise  between  the  z  used  here  and  the  z  formerly  employed. 

MTTt 

It  follows,*  since  z  =  e2(a,  that 

*>  =  Y\  17  =  3f» 

h2nz~2 


aj      aj2((z  -  z-1)2      mtt  (1  -  h'2nz~2}2      £(  (1  -  h2nz2}2 

where  h  =  e  ™  =  q. 

In  order  to  determine  the  constant  y,  it  follows,  when  we  expand 


urt 

z  =  e*"    and 


that 

and  consequently 


3-4 

We  note  that 


u2      3  •  4  a>2 

*  See  Schwarz,  Formeln  und  Lehrsdtze,  etc.,  p.  10. 


336  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  we  write  this  value  in  the  above  expression  for  @u,  we  have 

a)      u2       12  w2 


It  follows  that  * 


!     ,2          ^"g        2fr2n        _ 

w       12  w2      w2nf?[(l  -/i2")2 


or 


The  above  expression  for  §?w  is  not  unique,  since  the  period  2  a>  may  be 
chosen  in  an  indefinitely  large  number  of  ways. 

ART,  287.  Since  the  series  derived  in  the  last  Article  is  uniformly 
convergent,  we  may  integrate  term  by  term.  If  in  this  integration  we 
make  a  suitable  choice  of  the  constants,  we  again  have  a  convergent 
series. 

Multiplying  the  series  by  —  du,  it  follows  through  integration  that 

£u  =  -(u)=±-+  * 
a  u 

,  2h2nz~2  2h2n 


-h2nz~2       1  -h2n 


y    ^    2h2nz2  2h2n    n 

3i  ll  -h2nz2       1  -h2n\] 


where  the  choice  of  constants  has  been  such  that  the  constant  terms 
occurring  in  the  expressions  under  the  summation  signs,  when  expanded 
in  ascending  powers  of  u,  are  zero,  this  being  already  the  case  on  the 
left-hand  side  of  the  equation. 

The  above  formula  simplified  may  be  written  f 


,  2n~2 

tt  +  -_—  + 


If  with  Eisenstein  (loc.  tit.,  p.  215)  we  note  that 
-2  2h2nz~2  zh 


-  h2nz~2       l-h2nz~2  zh~n  -  z~lhn 

*  Schwarz,  loc.  cit.,  p.  8. 
t  Schwarz,  loc.  cit.,  p.  10. 


THE   WEIERSTKASSIAN   FUNCTIONS    f>u,  &,  <ru.  337 

and  further  that 


ni—  £ 

=  e    »,     zh~n  =  e2 

7T  Z  —  Z~l 


u'  2' 

it  is  seen  that  the  above  expression  may  be  written  * 

'  r  n=x  ( 

£U  =  ^(U)=1U+  JL  cot-^i  +  V    ]cot^-(u-  2nu)')- 
a  w          2aj\_       2a>       £?1  (        2a> 

n  =  x    , 

f+  T   ]cot-^-(u  +  2  ncof)  + 
n=l    '  2  ^ 

It  is  evident  that  the  constant  term  of  the  series  is  zero;  for  if  u  is  changed 
into  —  u,  the  right-hand  side  of  the  series  takes  its  opposite  value  and  is 
consequently  an  odd  function  of  u. 
If  u  is  increased  by  2  a>,  the  quantity  z  becomes  —  z,  for 


e 
It  follows  at  once  that 

-(u  +  2aj)=  2  r)  +  -  (u), 
a  a 

or  £(u  +  2w)=  ^u  +  2  fj. 

Writing  u  =  —  w  in  this  formula  we  have  (cf  .  Art.  258) 

tu  =  y, 

where  T?  is  finite  since  —  (a>)  is  finite. 
a 

We  saw  that 

p(-u  +  2,0)')=  <$u. 

Multiply  both  sides  of  this  formula  by  —  du  and  integrate.     It  follows  that 

-(u  +  2co')=  ^.(u)+2r)f, 
a  a 

or  £(u  +  2aj')=  £u  +  2  if, 

where  r[  is  the  constant  of  integration. 
Again  writing  u  =  —  a>f,  we  have 


By  interchanging  w  and  a)'  in  the  preceding  Article,  it  may  be  shown  that 


where  h0=  e   u/. 

*  Schwarz,  loc.  cit.,  p.  10;  see  also  Halphen,  Fonct.  Ellipt.,  t.  I,  p.  425;  Tannery  et 
Molk,  Fonct.  Ellipt.,  t.  II,  p.  237. 


338  THEORY   OF   ELLIPTIC    FUNCTIONS. 

From  the  formula 

w=--lgau,=  --f  £(U) 
du  G 


it  follows  that  (cf.  Art.  258) 

^u  =  °~(u)=-f\udu=  r**», 

a  J  J    VS 

where  %>u  =  s    and     du  =  --  ^4-  • 

VS 

The  constant  of  integration  on  the  right-hand  side  is  so  chosen  that  for 
sufficiently  large  values  of  s  the  series  on  the  right-han%ide  is  (cf.  Art.  279) 


p^ 

J    VS 


VS  L  24  s2      40  s3 

uiti 

ART.  288.     If  u  is  increased  by  2  a/,  then  2  =  e2""  becomes  z*h.     We 
consequently  have 


c(*  +  2o/)  =  c*  +  2  ?'=  att  +  *u  21  ^"^" :  + 

o»          2a)(hz  —  h    lz  n=1  x  -  ,«,       -*    - 

i   2  rX 

-(-  — « — . 

Comparing  this  formula  with  the  one  given  above  for  £u,  we  note  that 
here  under  the  first  summation  the  new  initial  term  is 

{-*  Q  - 

- — -,  which  may  be  written  =  ^ 1, 

1  —  z~2  z  —  z-1 

and  consequently  the  first  summation  is  transformed  into 

z  +  z-1  _  1 
z  —  z~l 

while  the  second  summation 

becomes 


We  further  note  that 

hz  +  h-^z~l      h2z2+  1  h2z2+ 


hz-h-tz-1      h2z2-l  l-h2z2 

It  follows  at  once  that 

C(u  +  2cw)=  C^  +  2^=  ^  +  ^^  +  -'i  -  ^L+i-  1  +  -^ 

w          2  (       1  -  h2z2  1  - 


THE   WEIERSTRASSIAN   FUNCTIONS    VU,  &,  au.  339 

so  that 


or  finally  (cf .  Art.  259) 

We  have  assumed  always  (Art.  86)  that  R  ( — )  >  0. 

\<Mj 

ART.  289.     Following  a  method  given  by  Forsyth  (Theory  of  Functions, 
p.  257)  we  offer  another  method  of  proving  the  formula  last  written. 

Consider   the   period-parallelogram   with  vertices  0,   2  a>,    2w',   2uj"  = 
2  oj  +  2  a/. 

By  sliding  this  parallelogram  parallel 
with  itself,  it  may  be  caused  to  take          u0^^'=u3  /        u0^ul=us 


a  position  such  that  for  all  points  on          /  /  / 

its  boundarv  and  within  the   interior        *  *—     — ^ — 


/  0  / 

(except  the  point  u  =  0)  the  function     /  / 

£u  has   the   character   of   an   integral  u0  w0+2 

function,  being  of  the  form  Fig.  71. 


It  follows  that 

lu  =  27ri, 


where  the  integration  has  been  taken  over  a  small  circle  about  u  =  0. 
Since  this  integral  is  the  same   as  that  taken  over  the  parallelogram 
,  we  have 


or 

2m  =          ^u  -  £(u  +  2a)')}du 


-  2  T'di^  +        32  7  <fa  =  -  4  T' 


'udu+  l£u<fuj 

*/  Us 


ART.  290.     If  we  multiply  by  dw  the  expression 
—  (u  +  2w)=-(u)+27/, 

C7  (7 

we  have  through  integration 

log  <J(M  +  2  o>)  =  log  <TU  +  2  TJW  +  c, 
or 


340  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  the  value  -co  is  given  to  u,  it  is  seen  that 

gC_  _  'e2^. 

We  consequently  have 

o(u  +  2a>)  =  —  e2^u+^a(u). 

If  —  u  is  written  for  u  in  this  formula,  we  have 

o(u  -  2w)=  -  e~2^u-^ou. 
Combining  these  formulas  into  one  formula,  we  may  write 

(A)  a(u±2a>)  =  -  e±2*(t«±«>  a(u). 
In  a  similar  manner  it  may  be  shown  that 

(B)  o(u  ±  2  a>')  =  -  e 


Further,  if  2  to  =  2  pco  +  2  qa*'  ',  where  p  and  q  are  positive  or  negative 
integers  (including  zero),  it  is  seen  that 

a(u  +  2w)  =  <7(u  +  2paj  +  2 
Writing 

2py  +  2qi)'=  2 
•it  follows  that 

a(u  +  2  a>) 

To  determine  the  constant  C,  write  u  =  —  co  +  v,  where  v  is  a  very  small 
quantity.     It  follows  that 


o(to  +  v)  =  —  Ce-27'"+2w<7(a)  —  v). 
If  we  develop  by  Taylor's  Theorem,  it  is  seen  that 
(C)          o(w  +  v)=  a(oj)  +  va'(a>)+  -   •   -    -  -  Ce-2w+2*vo 

Two  cases  are  possible: 

(1)  either  |  a(a>)     >  0,     or 

(2)  ,7(5)  |  -  0. 

In  the  first  case  we  have  by  writing  v  =  0, 

<r(5)=-  Ce~2^a(a)). 
It  follows  that  C  =  -  e2**, 

and  consequently 

a(u  +  2  w)  =  -  e2~r<(u+^o(u}. 

In  the  second  case  we  have  by  developing  both  sides  of  (C) 

<?'(£)+  (M)  -  C 
or  by  making  v  =  0, 

C  = 
It  follows  that 

<r(w  +  2  5)  =  ± 

according  as  we  have  case  (2)  or  case  (1)  respectively. 


THE    WEIERSTRASSIAN   FUNCTIONS    yu,  &,  <rii.  341 

The  quantity  a  (at)  vanishes  when  p  and  q  are  even  integers.     We  may 
therefore  write  the  general  formula 


a(u  +  2pa}  +  2qaj')  =  (- 
ART.  291.     We  derived  in  Art.  287  the  formula 


which  is  uniformly  convergent  within  the   period-parallelogram  (vertices 
excluded).     If  this  series  is  integrated  term  by  term,  it  follows  that 


log  „ 


When  u  =  0,  we  have  z  =  1,  so  that 

[log  <™]u=0=  C  +  log  sin  ^- 

L  2  wju= 


|f^  +((«=>)) 

L2^  Ju=o 

It  follows  that  * 


and 


where  ?i  =  2  wv. 

Writing  —  =  7,  it  is  seen  that 

CO 


1  —  h2nz~2  =  sin[(r  —  m)-]    _l  =  sm[(rn  —  r)~]  _1 
I  _  h'2n  h~n  —  hn  sin  TIT-    ~Z     ' 

2i 

with  a  similar  formula  for  -^-- 

It  follows  that  f 

(2)  ou  -  ^-'  ^ sm  v*  n sin[(nr~  rkig-^-  n ^i^±_ 

-  „         sinrir^-  „         sinnrr 

or 


n=i 


*  Compare  this  function  with  Eisenstein's  x-function,  loc.  cit.,  p.  216. 
t  Schwarz,  loc.  cit.,  p.  8.     Formulas  (2)  and  (3)  are  precisely  the  same  as  those 
derived  by  Jacobi  for  H(u)  [Werke,  I,  pp.  141-142]. 


342  THEORY   OF   ELLIPTIC    FUNCTIONS. 

The  formula  (2)  may  be  written 
(3)  ~- 


Since  2  aj  may  be  chosen  in  an  infinite  number  of  ways,  it  is  seen  that  au 
may  be  expressed  in  an  indefinite  number  of  ways  in  the  form  of  a  simply 
infinite  product.  Through  logarithmic  differentiation  of  formula  (3)  it 
follows  that  n=00 

TT  2r  h2nsm  2  vn 


10        , 
u  =  —  cot  vn  +  2  vv  H  -- 


2  a)  co  nr{  I  -  2  h2ncos  2  wr  +  h4" 

Noting  that 

-  L_  =   1   _|_  u  4-  U2+   .     .     .    +  Um+    -    •   :.  (  I  U  I    <    1), 

it  is  evident,  if  u  =  r(cos  6  +  i  sin  6),  that 

-  Srsintf  -  _  mv  2  rm  sin  m^ 
1  -  2  r  cos  ^  +  r2       ^ 

an  identity  which  is  true  for  complex  as  well  as  for  real  values  of  r. 
If  we  put  r  =  h2n,  we  have 


ii--'i  2 


and  consequently 


If  we  differentiate  with  regard  to  u,  we  have 
(A)  p 


4  OJ2  6t»  6t> 

The  right-hand  side  of  this  equation  is 

1       .     (7o       o    |     ^7j 

g?tt  =  —  +  **•  U    H~ 

while  the  expansion  of  cosec2  t  is 


By  equating  like  powers  of  u  on  either  side  of  (A),  we  have  * 


*  Harkness  and  Morley,  Theory  of  Functions,  p.  321;  Halphen,  Fonct.  Ellipt.,  t.  I, 
Chap.  13 


THE   WEIERSTRASSIAN   FUNCTIONS    <?u,  &,  <ru.  343 

ART.  292.     Homogeneity.  —  Write  the  functions  an,  £u,  pu  in  the  forms 
o»  -  o(u',  co,  a/)  =  a(u;  g2,  g3), 

It  follows  at  once  from  the  infinite  product  through  which  the  function  au 
is  defined  (Art.  272)  that 

where  A  is  any  quantity  real  or  imaginary. 
We  also  have 

and  consequently 


p(/w;  ko,  )*}')=  — 
In  the  formulas 


when  w  and  o>'  are  replaced  by  Xco  and  Aw',  w  becomes  Xwt  so  that  g^  and 
are  transformed  into 

2|    and    23. 
x4  /6 

It  is  also  seen  that 


The  above  formulas  are  particularly  useful  when  in  Volume  II  we  make  a 
distinction  between  the  real  and  imaginary  values  of  the  argument. 
ART.  293.     Degeneracy.  — When  a/=  oo,  we  saw  in  Art.  283  that 


We  further  have 

From  Chapter  I  we  have 


32  -o  m6       33  .5.7 


344  THEORY   OF   ELLIPTIC   FUNCTIONS. 

It  follows  that 

=  i/jELy     =  i  /  7T2  \3 

and  consequently  .  3        _     2_ 

The  discriminant  being  zero,  the  roots  of  the  polynomial 

4s3—  g2s  —  g3=  0  =  4(s  —  ei)(s  —  62)  (s  —  63) 
are  equal.     Further,  since 

61+62+63=0    and    ei>62>e3, 

the  quantity  e\  must  be  positive  and  63  negative. 

Two  cases  are  possible:  either  e2  coincides  with  63,  or  e2  coincides  with  e\. 
In  the  first  case:  62=  63=  —  \e\\  g2=3ei2,    g3=  6i3,    g3>  0; 


2 

We  also  have 


o'u        x       ,  nu    .If  x  \2 

u  =  —  =  —  cot  --  —  (  —  j  u, 

on       2co        2aj      3\2ajj 


l/Jrt*\2 

6\2^j  2<y  .    TTU 

GU  =   6bV       '   Sin 


In  the  second  case:  e2  =  e\=  —  \e^\   g%<.  0, 


92=  3e32,  ^3=  633;  k  =  I,  snu  =      ~_u,   K  =00,  w  =  oo. 

6  ~t~  e 

9  2,     where  ,  =  t« 


,    where 

,     ,          T? 

V--, 


in  2 

When  the  roots  of  the  polynomial 

4s3-  g2s  -  g3=  0 

are  equal,  it  may  be  shown  directly  that  the  values  of  s  =  @u  derived 
from  the  integral 

a)  u 


V4s3-  g2s-  g3 
agree  with  the  results  above- 


THE   WEIEKSTRASSIAN   FUNCTIONS    &u,  &,  <m.  345 

When  both  periods  are  infinite,  then  g2  =  0  =  g$  and  e\  =  0  =  e<i  =  €3. 
The  integral  (1)  becomes 


u=         -=,     or     5  =       = 


-  r  d±_ 

Js  vTs* 


-  ,     ou  =  u. 
u 


EXAMPLES 

1.    By  making  a/  =  oo  in  the  formula 

~  +  2 


derive  the  results  of  Arts.  283  et  seq.  (Halphen,  loc.  tit.,  Chap.  13). 

*.  K  /-  f     ds     , 

Jo    V±  s3  -  a,s 


show  that 


3.    If  F(f)  is  any  rational  function  of  t  =  c  ™  ,  such  that  F(0)  =  0  =  F(oc), 
show  that 


n=l  n=l 

is  a  one-valued  doubly  period  function  of  u. 


CHAPTER   XVI 
THE    ADDITION-THEOREMS 

ARTICLE  294.  It  is  the  purport  of  this  treatise  to  consider  as  far  as 
possible  the  ultimate  meaning  of  the  functions  which  have  been  intro 
duced.  The  simplest  funct'onal  elements  have  been  found  in  the  Jacobi 
Theta-functions  which  are  made  the  foundation  of  the  theory.  It  is 
therefore  natural  first  to  develop  the  addition -theorems  from  this  stand 
point. 

We  have  seen  in  Art.  90  that  there  exists  a  linear  homogeneous  equation 
with  coefficients  that  are  independent  of  the  variable  among  any  n  +  1 
intermediary  functions  <&(u)  of  the  nth  order,  which  have  the  same  periods. 
We  may  next  make  an  application  of  this  theorem  for  the  case  n  =  2. 
If  in  Art.  87  we  write 

a  =  2K,     6  =  2  iK',     n  =  2, 
it  follows  that 


(I) 


2iK')  =  e    K^ 


Among  any  three  functions  of  the  second  order  which  satisfy  these  func 
tional  equations  there  must   exist  a  linear   homogeneous    equation  with 
coefficients  that  are  independent  of  the  variable.* 
Three  such  functions  are 

@2 (w),     H2(w)     and     ®(u  -  v)@(u  +  v), 

where  v  is  an  arbitrary  parameter. 
It  follows  that 

C®(u  +  v)®(u  -v)  +  Ci02(w)+  C2H2(w)=  0, 

where  the  C's  are  quantities  independent  of  u.      The  C's  may,  however, 
be  functions  of  v. 

None  of  these  quantities  can  be  zero;  if,  for  example,  C  =  0,  we  would 
have 

5^  =  Constant, 

8(«) 

which  is  not  true. 

*  See    Hermite  in  Serret's  Calcul,  t.  II,  p.  797;  and  Koenigsberger,  Elliptische 
Functionen,  p.  368. 

346 


THE   ADDITION-THEOREMS.  347 

Writing 


we  have 

B(u  +  v)®(u  ~v)  =  f(v)  ®2(u)  +  g(v)  H2(u). 

If  we  consider  f(v)  G2  (u)  +  g  (v)  H2  (u)  as  a  function  of  v,  say  ¥(v),  we 
have 


It  follows  that 

¥(i?  +  2K)=  ¥(r) 
and 

2;rt 

-rir*  *V(i'), 

from  which  it  is  seen  that  "^(v)  satisfies  the  functional  equations  (I). 
If  we  write  v  +  2  K  in  the  equation 

(II)  0(W  +  V)  0(u  -  v)  =  f(v)  02(w)  +  gr(v)  H2(u), 

we  have 

0(t*  +  v)Q(u  -  v)  =  f(v  +  2K)02(^)+  g(v  +  2K)H2(u); 
and  consequently  through  subtraction  it  follows  that 

[f(v  +  2  K)  -  f(v)]  02(M)  +  [gr(y  +  2  K)  -  </(iO]  H2(M)  -  0. 
As  this  relation  is  true  for  all  values  of  u,  we  must  have 

/(•  +  2JO-M 

0(0  +  2/0=0(1;). 

On  the  other  hand,  if  in  the  equation  (II)  we  write  v  +  2  i7£'  for  v,  we 
have  hi  a  similar  manner 


g(v  +  2  iK')=e~* 

It  follows  that  /(v)   and  g(v)   satisfy  the  functional  equations  (I)    that 
were  satisfied  by  02(u)  and  H2(w). 
We  thus  have  the  following  relations: 


g(v) 

where  a,  /?,  7-,  ^  are  constants. 

When  these  relations  are  written  in  the  equation  above,  we  have 

(1)    0(w  +  v)B(u  -  r)= 


348  THEOKY   OF   ELLIPTIC    FUNCTIONS. 

To  determine  the  constants  a,  ft,  ?,  d,  write  v  =  0.     We  then  have 


G2o)[i  -  /?e2(o)]=  £02(o)  n2(u), 

a  relation  which  can  exist  only  if 

1  -  /?@2(0)  =0     and     £B2(0)  =  0. 


We  thus  have 

1 


and  *- 


If  next  we  write  w  =  0  in  the  above  equation,  we  have  a  =  0.  To  deter 
mine  7-,  we  write  the  values  of  a,  ft,  d  just  found,  in  (1),  then  write  u  = 
v  +  iK'  and  note  that  ®(iK')  =  0.  It  follows  that 


@2(0) 

These   values  of  a,  ft,  f,  d  when  written  in  the  equation  (1)  give  us  the 
formula 

v)®(u  -  v)=S2(v)(d2(u)-R2 


which  is  fundamental  in  the  Jacobi  theory  (see  Jacobi,  Werke,  I,  p.  227. 
formula  20). 

ART.  295.     We  introduced  in  Art.  208  the  followin    notation: 


H(2  Ku)  = 


We  also  saw  in  Art.  215  that 


j_  =Qi(Q) 

Vk      Hi(0) 
/r,  ^  0(0) 


and  in  Art.  217  that 


The  addition  formula  above  for  the  function  0  may  be  written 
(1)  &02&o(u  +  v)  &0(u  -v}= 


THE   ADDITIOX-THEOKEMS.  349 

if  in  the  original  formula  we  write  2  Ku  for  u  and  2  Kv  instead  of  v.     In  a 
similar  manner  we  may  derive 

(2)  #2 

(3)  & 

(4) 


All  four  of  the  above  formulas  were  also  derived  in  the  table  (C)  of  Art.  211. 
ART.  296.     If  we  divide  equation  (2)  above  by  (1)  we  have 


that  is, 


#0 


or 

sn[2K(u  +  v}~\=  sn  ^  ^u  cn  ^  ^l  dn2  Kv  +  cn2  Ku  dn  2  Ku  sn  2  Kv 

If  we  divide  the  equation  (3)  by  (1)  we  have 

cn  2  Ku  cn  2  Kv  -  sn  2  Ku  sn  2  Kv  dn  2  Ku  dn  2  Kv 


1  -  k2sn22Kusn22Kv 
and  similarly  when  (4)  is  divided  by  (1)  we  have 

dn\2K(i  +  01  =  dn 2  Ku  dn  2  Kv  ~  k2sn2 2  Ku sn  2  Kv cn  2  Ku cn  2  Kv 

1  -k2sn22Kusn22Kv 

If  we  write  u  and  v  for  2  Ku  and  2  Kv,  we  have 

sn(u  +  v}  =  snucnvdnv  +  cnudnusnv 

1  —  k2sn2usn2v 
Further,  since  , 

—-snu  =  cn  u  dn  u, 
du 

it  follows  that  dsnv^          d  sn  u 

sn  u f-  sn  v 


.    r>  •>    o 

1  —  K~sn~u  sn^v 

We  have  thus  shown  that  sn(u  +  v)  is  a  rational  function  of  snu,  snv  and 
the  first  derivatives  of  these  functions  (see  Art.  158). 

Remark.  —  If  for  brevity  in  the  formula  above  we  put  sn  u  =  s,  snv  =  s'; 
cnu  =  c,  cnv  =  c';  dnu  =  d,  dnv  =  d',  it  becomes 


350  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  further  have 

cn2(u  +  v)=l-  sn2(u  +  v)=  ^ 


(1  -  k2s2sf2)2 
(ccf-  ss'dd'}2 


so  that 

Cn(u  +  V)  =  ±c-^-ss'dd' 


Writing  v  =  0,  and  consequently  s'  =  0  and  c'  =  1  in  this  formula  it  fol 
lows  that  en  u  =  ±  c,  so  that  the  positive  sign  must  be  taken.  We  may 
derive  the  formula  for  dn(u  +  v)  in  a  similar  manner. 

ART.  297.     Addition-theorem  for  the  elliptic  integrals  of  the  second  kind.  — 
From  the  formula 

02(0)  ®(u  +  v)  ®(u  -v)=  ®2(u)  ®2(v)  - 
we  have  at  once 


This  formula  differentiated  logarithmically  with  respect  to  u  and  v  respec 
tively  becomes 

®'(u  +  v)    .  ®'(u  -  v)  _  2®f(u)  ==_  2  k2sn  u  cnudnu  sn2v 
®(u  +  v)       ®(u-v)          9(u)  ~  1  -  k2sn2u  sn2v 

®'(u  +  v)      ®'(u  -  v)       0  Qr  (v)  =      2  k2sn  v  en  v  dn  v  sn2u 
®(u  +  v)       ®(u  -  v)  ~    "  ®(v)  ~  1  -  k2sn2usn2v 

Through  addition  we  have 


Since 

it  fpllows  that 

Z(u  +  v)  =  Z(u)  +  Z(v)  —  k2sn  usnv  sn(u  +  v). 

Noting  that 

Z(u)=E(u)-uL 

A 

we  also  have 

E(u  +  v)=  E(u)  +  E(v)  -  k2  sn  u  sn  v  sn(u  + 


THE   ADDITION-THEOREMS.  351 


ADDITION-THEOREMS  FOR  THE  WEIERSTRASSIAN  FUNCTIONS. 

ART.  298.     The  addition-theorem  for  the  ^function  may  be  derived  as 
follows:  We  note  that  the  difference 


is  a  one-valued  doubly  periodic  function  which  becomes  infinite  of  the 
second  order  at  the  origin  and  the  congruent  points.  For  all  other  points 
this  difference  is  finite.  The  points  u=±v  +  2luaj  +  2  //&/(/*,  //  integers) 
are  the  zeros  of  the  function  <$u  —  pv. 

Another  function  which  has  the  same  zeros  is 


0U 

Further,  since 


a(u  +  2  w)  =  -  e2  '("+">  au,     o(u  +  2  a/)  =  -  e2  »'(«+"')  <ru, 
it  follows  that 


and  <j)(u  +  2co') 

We  note  that  the  functions  $(11)  and  pu  —  $>v  have  the  same  periods. 

The  developments  of  these  functions  in  the  neighborhood  of  the  origin 
are 

pu  -  &v  =  —  -  §>y  +  (O2)), 


It  is  further  seen  that  the  function 


au 


is  doubly  periodic  and  becomes  infinite  in  the  same  manner  and  at  the 
same  points  as  <@u  —  $rv. 
Other  developments  are 

fy 

o(v  +  u}  =  av  +  ua'v  +  —  o"v  +  •  •  -  , 
Z 

G(—  v  +  u)  =  —  av  +  ua'v  -  —  a"v  +  •  •  •   , 
,    M  =  a2v  +  [ov<j»y  - 


02(V)U2+((U*}) 

or 


U  0V 


352  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

Since 


o£v 
we  may  write 

#i(t*)--5- 

This  value  substituted  in 


—  ®v  —  <f>i(u)  =  .Constant, 


shows  that  the  constant  is  zero. 
We  therefore  have 


a  formula  of  great  elegance  and  importance.* 

ART.  299.     If  the  formula  above  be  differentiated  logarithmically  respec 
tively  with  regard  to  u  and  v,  we  have 

(A)  -  (u  +  v)  +  —  (u  -  v)  -  2  -  (u)  = 
o  a  o 

and 

(B)  -  (u  +  v)--  (u  -v)-2~(v)  = 

o  o  a  <@u  — 

Through  the  addition  and   subtraction  of   formulas  (A)  and  (B)  are  de 
rived  the  formulas  f 


(C)  s  (M 

and 


•o  a  a  Z  <@u  —  %>v 


<      — 


These  formulas  are  the  addition-theorems  for  the  function  —  (u)  =  £(u). 

a 

Compare  them  with  those  given  in  Art.  297.     The  function  %u  does  not 
have  an  algebraic  addition-theorem.  J 

If  we  differentiate  again  the  formula  (C)  with  respect  to  u  and  v,  we  have 


(E) 


=  _ 

2  (pu  -  gw)2 

and 

(F) 


*  See  Schwarz,  loc.  cit.,  p.  13. 

t  Schwarz,  loc.  cit.,  p.  13. 

j  Daniels,  Amer.  Journ.  Math.,  Vol.  VI,  p.  268. 


THE   ADDITION-THEOREMS.  353 

It  follows,  since 


-  g2  &ti  -  g3     and     p"u  =  6  g?u  -  J  g2, 
that  the  formula  (E)  becomes 

(E')  g>(w  ±  v)  =  ou  -  ^U 


while  formula  (F)  may  be  written 
(F')    o(u  ±  v)  =  &v  +  fcM 


-  pi;)2 

ire  /love  thus   expressed  $>(u  ±  v)  rationally  through  $>u,  pv,  p'u,  $>'v  (see 
again  Art.  158). 

ART.  300.     Through   the   addition  of   the   formulas   (E')  and    (Fr)  we 
have 

(G)  &(u  ±  v)  =  2(VU<?V 


2(<pu  -  yv)2 

The  function  p(u  +  v)  is  only  infinite  if  u  is  equal  or  congruent  to  —  v. 
Since  #>w  is  finite  at  this  point,  it  follows  from  the  formula 


that  the  partial  differential  quotient  which  appears  on  the  right-hand 
side  must  be  infinite  for  the  value  u  =  —  v. 
To  observe  the  nature  of  this  infinity,  write 


u  =  —  v  +  h. 
It  follows  that 


and  that 


du  (  yu  — 


Noting  these  results  we  may  obtain  another  formula  for  p(u  ±  v)  as  follows: 
The  function 


is  one- valued  and  doubly  periodic.  It  is  also  finite  at  the  point  u  =  —  v 
and  the  congruent  points.  We  further  note  that  this  function  remains 
finite  at  the  point  u  =  +v.  At  the  point  u  =  0  the  function  becomes 

infinite  as —  •     If  then  we  add  to  the  above  expression  the  function 

u2 
v?w,  we  have  a  doubly  periodic  function  which  remains  finite  everywhere 


354 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


in  the  finite  portion  of  the  plane  and  is  therefore  (see  Art.  83)  a  constant. 
It  is  easily  shown  that  this  constant  is  —  <@v. 
We  may  consequently  write 


v      1  \v'u  =F  6/vl2 

u  ±  v)  =  -  • -!-*—     -  pu  -  gw. 

4  I  pu  -  $>v  J 


ART.  301.     If  in  the  formula  just  written  we  put  u  +  v  f  or  u  and  —  v 
for  v,  we  have 


It  follows  that 


+^ 

-  gw  J 


/fri  +  v)+  p'y 
(u  +  v)-  §w 


If  both  sides  of  this  equation  are  developed  in  powers  of  u,  it  is  seen  that 
the  negative  sign  must  be  used. 

In  determinant  form  this  formula  may  be  written 


1,     pu, 
1,     gw, 


=  0. 


By  differentiating  with  regard  to  v  the  formula 


/  \  19    fp'u  -F  (ip'v 

(^  -t  v)=  &u  -     _Ji  -  L_ 

2  6w     ?w  —    ?v 


we  have 


2  du  dv 


Remark.  —  If  in  the  formula   (Fx)  of  Art.  299  we  write  w  in  the  place 
of  v  and  observe  that 


-03=4 


0, 


it  is  seen  that 
or 

From  the  relation 

it  follows  that 

and  consequently  that 


=  6 


THE   ADDITION-THEOREMS.  355 

Further,  since 


-  e2)(<?u  -  e3), 
and  therefore  also 
tf'u  =  2[(pu  -  ei)(&u  -  e2)  +  (pu  -  ei)(pu  -  e3)  +  (pu  -  e2)(pu-e3)], 

it  follows  that 

$>"a)  =  2(ei-  e2)(ei-  e3). 

We  consequently  have 


$>u  -  el 
and  similarly  (u  ±  ^  _  ^  =  (e2-  ejfa-  e3)  , 


ART.  302.     The  reciprocal  of  formula  (G),  Art.  300,  is 


±  v)       2($>u&v  -  J  g2)  (pu  +  &v)  -  g3 


Noting  that 

(&'uY=  4  $Pu  -  g2pu  -  g3    and    (p'v)2  =  4  g>3v  -  g2?v  -  g3, 
it  is  seen  that 

gw)  -  Q3]2  -  [4  g?3?^  -  g2<?u  -  g3]  [4  $3v  -  ^2^?'  -  fl'a] 


and  consequently 


ff(u  ±  v) 
If  we  write  *£  =  v,  we  have 
(2i0= 


4  p3^ 
It  also  follows  that 

-  3    4u 


,  A 

ttj  —  g/it  — 


-  g3 


From  the  formula  just  written  we  have 


356  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

Integrating  we  have      /  j  \    A 

^(2u)=2?-(u)+1-f\ogv'u  +  C 
G  o  2  du 

=  2^)+i^£  +  C. 
a  2  p'u 

Developing  both  sides  of  this  expression  in  ascending  powers  of  u,  it  is 
seen  that  the  constant  (7  =  0. 
We  therefore  have 


This  formula  multiplied  by  2  du  and  integrated  becomes 

log  a(2u)=  4  log  au  +  log  <@fu  +  log  c, 

so  that 

a(2u)=  c(au)*p'u. 

It  follows  that 


from  which  it  is  seen  that  c  =  —  1  and  consequently 

o(2  u) 


ART.  303.  Historical. — It  was  known  through  the  works  of  Fagnano, 
Landen,  Jacob  Bernoulli  and  others  that  the  expressions  for  sin  (a  +  /?), 
sin  (a  —  /?,)  etc.,  gave  a  means  of  adding  or  subtracting  the  arcs  of  circles, 
and  that  between  the  limits  of  two  integrals  that  express  lengths  of  arc 
of  a  lemniscate  an  algebraic  relation  exists,  such  that  the  arc  of  a  lemnis- 
cate  although  a  transcendent  of  higher  order,  may  be  doubled  or  halved  just 
as  the  arc  of  a  circle  by  means  of  geometric  construction. 

It  was  natural  to  inquire  if  the  ellipse,  hyperbola,  etc.,  did  not  have 
similar  properties.  Investigating  such  properties  Euler  made  the  remark 
able  discovery  of  the  addition-theorem  of  elliptic  integrals  (see  Nov.  Comm. 
Petrop.  VI,  pp.  58-84,  1761;  and  VII,  p..  3;  VIII,  p.  83). 

Euler  shows  that  if 


f  #    i  r  #  _  r- 

/         /  I         /  —    I 

«/o    v  R(£)      Jo    v  R(fz}      J® 


where  R(£)  is  a  rational  integral  function  of  the  fourth  degree  in  f,  there 
exists  among  the  upper  limits  x,  y,  a  of  the  integrals  an  algebraic  relation 
which  is  the  addition-theorem  of  the  arcs  of  an  ellipse  and  is  the  algebraic 
solution  (cf.  again  Euler,  Nov.  Comm.  Vol.  X,  pp.  3-56)  of  the  differential 

ecLuation 


Euler  states  that  the  above  results  were  obtained  not  by  any  method,  but 
potius  tentando,  vel  divinando,  and  suggested  that  mathematicians  seek  a 


THE    ADDITION-THEOREMS.  357 

direct  proof.  The  numerous  discoveries  of  Euler  are  systematized  in  his 
work  Institutiones  Calculi  Integralis,  Vol.  I,  Sectio  Secunda,  Caput  VI. 

The  fourth  volume  (p.  446)  contains  an  extension  of  the  addition-theorem 
to  the  integrals  of  the  second  and  third  kinds.  This  work  must  there 
fore  have  proved  of  great  value  to  Legendre  in  the  development  of  his 
theory.  In  every  case  geometrical  application  of  the  formulas  was  made 
by  Euler  for  the  comparison  of  elliptic  arcs. 

The  suggestion  made  by  Euler  that  one  should  find  a  direct  method  of 
integrating  the  differential  equation  proposed  by  him,  was  carried  out  by 
Lagrange,  who  by  direct  methods  integrated  this  equation  and  in  a  manner 
which  elicited  the  great  admiration  of  Euler  (see  Miscell.  Taurin.  IV,  1768; 
or  Serret's  (Euvres  de  Lagrange,  t.  II,  p.  533). 

The  addition-theorem  for  elliptic  integrals  gave  to  the  elliptic  functions 
a  meaning  in  higher  analysis  similar  to  that  which  the  cyclometric  and 
logarithmic  functions  had  enjoyed  for  a  long  time. 

ART.  304.  We  may  consider  next  some  of  the  general  investigations 
which  led  Euler  to  the  discovery  of  the  addition-theorem  and  then  give 
his  solution  and  the  one  of  Lagrange. 

If  we  differentiate  the  equation 

(I)  Ax2+  2  Rxy  +  Cy2  +  2  Dx  +  2  Ey  +  F  =  0, 
we  have 

(II)  (Bx  +  Cy  +  E)  dy  +  (Ax  +  By  +  D)dx  =  0. 

From  (I)  we  have 

x  =  -  %L±D.  ±  4 

^rl  ^1 

Bx  +  E 


±  i  V(Bx  +  E)2  -  (Ax2  +  2Dx  +  F)C. 

These  values  substituted  hi  (II)  give 

(III)  !fe_  +  -4±L=  =  0, 

where 

F(x)  =  (Bx  +  E)2-(Ax2+  2  Dx  +  F)C, 

G(y)  -  (J?i/  +  D)2-  (C?/2+  2  Ei/  +  F)A. 

If  .4  =  C  and  D  =  E,  then  G(i/)  becomes  F(*/).     The  differential  equation 
(III)  becomes  thereby 


VF(x) 
and  its  algebraic  integral  is 

(F)  A(x2  4-  y2}  +  2  Bxy  +  2  D(x  +  y)  +  F  =  0. 


358  THEORY   OF   ELLIPTIC    FUNCTIONS. 

Suppose  next  that  R(x)  =  ax2  +  2  bx  +  c  is  given  and  it  is  required  to 
find  the  integral  of  dx 


We  must  so  determine  the  constants  A,  B,  F,  D  that 

ax2+  2bx  +  c  =(Bx  +  D)2-  A(Ax2+  2  Dx  +  F). 

By  equating  like  powers  of  x,  we  have  three  relations  existing  among  the 
four  quantities  A,  B,F,  D.  We  may  therefore  determine  B,  F,  D  in  terms 
of  A. 

It  follows  that  the  differential  equation 

VR(X)     VR^) 

is  always  integrable  through  an  algebraic  equation  (F)  of  the  second 
degree  which  is  symmetric  in  x  and  y  and  contains  an  arbitrary  constant  A . 
By  the  comparison  of  this  algebraic  equation  with  a  transcendental  equa 
tion  which  we  shall  determine  later,  we  derive  the  associated  addition- 
theorem. 

If  further  we  observe  that  -  aR(x)  =  (b2-  ac)  fl  -  /  ax  +  b  VI  and  put 

ax  +  b   =  z,  then 
Vb2-ac  

r  -7=r  =  u>  saY> 

Jx*.  \/R(xn) 


becomes,  if  we  take  the  minus  sign  with  the  root, 

au  =  I  dz  where     s2=  I  -  z2, 

Jz0,s0Vl   -  Z2 

or  dz___ 

du 

If  s  is  not  a  one-valued  function  of  z,  there  must  be  a  second  branch  of  the 
function,  which  in  the  Riemann  surface  is  represented  on  a  second  leaf, 
so  that  if  zi  represents  the  variable  z  in  this  leaf,  we  have 


du 

ART.  305.     It  is  evident  that  we  may  write  the  differential  equation 

dx . dy =  Q 

Vox2  +  2  bx  +  c      Vay2  +  2  by  +  c 

0, 


in  the  form  ^  dj] 


Vl  -  £2      Vl  -  f)2 
or 


THE   ADDITION-THEOREMS.  359 

If  r,  is  a  function  of  £  which  satisfies  this  differential  equation,  then  is 


where  C  is  the  constant  of  integration.     Integrating  by  parts  we  have 
at  once 


c  =  ~ 


V 1  -  if      v  1  -  £2J 

or  ^ / — 

This  is  the  algebraic  integral  of  the  differential  equation  and  corresponds 
to  the  integral  (I')  of  Art.  304,  which  latter  equation  was  derived  through 
experimenting  by  Euler.  To  determine  the  corresponding  transcendental 

integral  write 


(1)  u=       '—-==,     where  a  =Vl  -  ^,     and 

•/ft  1  V  1  -  ^ 


(2) 


/^n  *  rl  

v  =  I  '       ,     where  -  =Vl  —  -n2. 

Jo,  i  \  1  -  r/2 


It  follows  that  £  =  sin  u  and  y  =  sin  v. 
The  differential  equation 


VI  -  $2       Vl  -  rf 

becomes  du  +  dv  =  0. 

We  therefore  have 


(**>•       d;        +    A.t       ^        = 

Jo,  i  \/l  -  c2      Jo,  i  \  1  -  r/2 


or  w  +  v  =  c, 

which  is  the  transcendental  integral  of  the  above  differential  equation. 
We  so  determine  the  constant  C  in  the  algebraic  integral 


that   for  c  =  0,   o  =  +l   the   variable    TJ  takes   the  definite  value  rl0.     It- 
follows  at  once  that 

C  =  rl0. 

When  the  values  £  =  0,  a  =  + 1  are  written  in  the  upper  limit   of   the 
integral  (1),  it  is  seen  that 

u  =  0, 

and  since  u  +  v  =  c,  it  follows  that 


fjo.Vl^      dr) 
c  =  I  ' 

•/  V     -     2 


Ctl  -  ry 

7j0  =  sin  c  -  sin  (t*  +  v). 


360  THEORY   OF   ELLIPTIC   FUNCTIONS. 

On  the  other  hand,  since 

we  have  gin  (^  _j_  v)  =  sjn  u  cos  v  4.  sjn  v  cos  Wj 

which  is  the  addition-theorem  for  the  sine-function. 

ART.  306.     In  a  similar  manner   Euler  derived  the  addition-theorem 
for  sn  u  as  follows. 

Suppose  we  have  given  the  quadratic  equation 

(I)  Ay2  +  2  By  +  C  =  F(£,  T?)  =  0, 

where  A  =  a0£2+  2ai£  +  a2, 

J5  -J 


By  arranging  the  terms  according  to  powers  of  £,  the  same  quadratic  equa 
tion  may  be  written 

A'$2  +  2  £'£  +  C'  =  F(?,  T?)  =  0, 
where  A'  =arf+  260>?  +  co, 

5'=  a^2+  2  6^  +  ci, 
C'  =  a2>?2  +  2  62^  +  c2. 
Differentiating  (I)  we  have 

dr  jf.    .    dr    -,          ^ 
TJ^  +  —  ^  =  0, 

d£  dri 

or  (A7  +  B')  d£  +  (Ar;  +  B)di)  =  0. 

It  follows  *  at  once  that 

=  0. 


Ai)+B     A' 

On  the  other  hand  we  have 


or  At]  +  5  =      B2-  AC, 

where  both  signs  may  be  associated  with  the  root;  and  similarly  we  have 

A'£  +  B'= 
We  thus  derive  t  the  equation 

(II) 

VB2-AC      VB'2-A'C 

*  See  Euler,  loc.  cit.,  or  Enneper,  Elliptische  Funktionen,  p.  186. 

t  See  Euler,  Institutiones  Calc.  Int.,  Vol.  I,  Sectio  Secunda,  Caput  VI;  or  Lagrange 
(1766-69),  (Euvres  (Serret,  Paris,  1868),  t.  II,  p.  533.  Halphen  (Fonct.  Ellipt.,Vo\.  II, 
Chap.  IX)  calls  such  an  equation  an  Euler-equation  and  remarks  that  by  the  dis 
covery  of  the  general  integral  of  this  equation  "Euler  sowed  the  first  germ  of  the  theory 
of  elliptic  functions  "  (in  1761). 


THE   ADDITION-THEOREMS.  361 

or 


2+  2&!*  4-  62)2-  (a0£2+  2  a,*  +  a2)(c<*2  +  2c^  +  c2) 

+    ,  ^  =0. 

V(ai^2+26i7?  +  c1)2-(a0^2+260>?+co)(a27/2  +  2627?  +  C2) 

If  we  put  ai=&0,     «2=  c0,     b2=  ci, 

the  expressions  under  the  roots  take  the  same  form,  while  equation  (I) 
becomes  * 

(I')      a0cV+  2  &o^(£+  T?)  +  c0  (£*  +  >?2)  +  4  fc^  +  2  ^(£+7?)  +  c2=  0. 
If  the  differential  equation  which  we  wish  to  integrate  is 

(III)  -^L=  +  -^=.  -  0, 


where  72(0=  P0*4  +  Pi*3  +  P2^2+  ^3^  +  P*,  we  may  make  this  equation 
identical  with  (II)  by  writing 

B2-  AC  =  R(?), 

or     (&0£2+  2  61^  +  62)2-(oo^+  2a^ 
We  therefore  have  the  conditions 


PI= 

P2=  260^>2+  ±bi2—  aQc2—  4aiCi—  a2c0, 
P3=  46i&2—  2aic2  —  2a2ci, 
P4=  622-  a2c2. 

Thus  in  addition  to  the  three  conditions  a\  =  bo,  a2  =  CQ,  62  =  c\  we  have 
the  above  five  conditions  among  the  nine  quantities  aQ,  bo,  c0,  ai,  61,  ci, 
«2,  b2,  c2. 

It  is  evident  that  when  these  conditions  have  been  satisfied  there  remains 
an  arbitrary  constant  in  the  equation  (I'),  which  equation  is  the  algebraic 
integral  of  (III). 

ART.  307.     In  particular  let  the  equation  (III)  have  the  form 

(III)'  ^  +  drl  =  0. 


Noting  from  above  that  ai  =  bQ,     a2  =  c0,     b2  =  ci,  we  have 

&o2—  «  Oc0=  k2, 
(2  61-  c0)&o-  a-oci=  0, 
4612-a0c2-co2-260ci=-  (1  +  k2), 
(2bi-  CQ)CI-  b0c2=  0, 

Ci2-  C0C2=    1. 

*  See  Cayley,  ?oc.  ci/.,  p.  341. 


362  THEORY   OF   ELLIPTIC   FUNCTIONS. 

We  observe  that  (III7)  remains  unchanged  if  £  and  T?  are  replaced  by  —  c 
and  —i).  It  follows  that  (I')  must  remain  unaltered  by  this  transforma 
tion.  We  must  therefore  have 

b0=Q,     ci=0. 
The  relations  just  written  are  consequently 

—  coC2=  1,     4&i2  —  &oC2-  Co2+  1  +  &2  =  0,     —  aoCo=  k2, 
or 

1  k2 


co2 


Writing  these  values  in  equation  (F)  we  have 


Co         CQ  CQ 

or 

[1  +  &2£V+  c02(£2  +  r)2)]2=  4[/c2-(l 

Arranged  in  powers  of  —  ,  this  equation  is 
Co 

2(1    +  fc2£V)(£2   +   7?2)-   4(1 


co4 


=  7]  Vl  - 

c0  ~  1 


or 


which  is  the  algebraic  integral  of  (III').  After  deriving  the  transcen 
dental  integral  Euler  proceeded  to  the  addition-theorem  in  practically  the 
same  manner  as  is  given  in  the  next  Article. 

ART.  308.     Professor    Darboux  *    proceeded    to    the   above  algebraic 
integral  as  follows:    He  assumed  that 


or      .  „ 


(i) 


where  Z(£)     (1  - 

and  required  that  £  be  determined  as  a  function  of  u. 
He  further  introduced  an  auxiliary  variable  v,  such  that 


<*  '- 


*  Darboux,  Ann.  de  I'Ecole  Norm.,  IV,  p.  85  (1867). 


THE   ADDITION-THEOREMS.  363 

We  therefore  have  from  (III') 

du  +  dv  =  0, 

u  +  v  =  c,     v  =  —u  +  c, 
where  c  is  a  constant. 

It  follows  that  j_  - 

23=-  Vfi(,), 

du 
so  that  £  and  >?  are  functions  of  u,  both  being  integrals  of  the  equation 


We  next  form  d          1   Z'(~)    d         1 


dw      2 

--(1  +  fc2)£  + 
and 


We  have  immediately 


Through  division  it  follows  that 


_ 

2      9Z^-J   #  x  - 
~  -+Cj 

?M         du 


du       *  du 
This  expression,  when  integrated,  becomes 


du  =  g 

~ 


where  C  is  a  constant. 
Further,  since 


we  have  at  once 


which  is  the  algebraic  integral  of  (III')- 


364  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  addition-theorem  may  be  derived  as  follows:     If  in  the  relation 

u  +  v  =  c 
we  write  for  u  and  v  their  values  from  (i)  and  (ii),  wre  have 

(N) 


AXz(g)     ft      +    A 
Jo,i          VZ(£)       Jo,i 


This  is  also  an  integral  of  (III')  but  in  transcendental  form. 

Suppose  next  that  y  becomes  TJO  for  the  values  £  =  0,  \/Z(£)  =  1.     It 
follows  from  (M)  that 


and  from  (N)  that 


If  we  write 

£  =  sn  u,  7)  =  sn  v, 


—  £ 2  =  en  u,         \/l  —  rj2  =  en  v, 
dnu,     VI  —  k2j]2  =  dnv, 


then  from  (P)  we  have 

7)0  =  snc. 

But  since  c  =  u  +  v  and  also  T?O=  C,  the'  equation  (M)  may  be  written 

snvcnudnu  +  snucnv  dnv          ,         \ 

1 =  sn(u  -f-  v). 

Write 

D  =  1  —  k2sn2u  sn2v 

and  note,  since  1  =  sn2u  +  cn2u,  that 

D  =  cn2u  4-  sn2u  dn2v  =  DI,  say, 
and 

and  also  that 

D2=  DiD2. 
It  follows  that 

Z)2  —  (sn  u  cnv  dnv  +  snvcnudnu}2 

__        ^__^^_^_____________^__^_^__— _  y 

or  (cf.  Art.  296) 

C     I  y\  =  c^^  cnv  —  snu  snv  dnu  dnv  t 
1  —  k2sn2u  sn2v 


THE   ADDITION-THEOREMS.  365 

Similarly,  if  we  note  that 

D  =  dn2u  +  k2sn2ucn2v  =  D3 
and  that 

D  =  dn2v  +  k2sn2vcn2u  =  Z)4, 
we  may  derive  from 

dn?(u  +  r)=  1  -  k2sn?(u  +  v) 
the  formula 

dn(u  +  r)  =  ^n  u  ^n  r  ~  ^2sn  u  cn  u  snv  cnv 
1  —  k'2sn2u  sn2v 

ART.  309.     A  direct  process  for  finding  the  algebraic  integral  was  given 
by  Lagrange  as  follows: 

For  brevity  write    X  =  a  +  bx  +  ex2  +  dx3  +  ex4, 

Y  =  a  +  by  +  cy2  +  dy3  +  ey4. 

The  differential  equation  to  be  integrated  is  of  the  form 
(I)  ^L  +  -^L  =  0. 

Vx    VY 

Considering  x  and  y  as  functions  of  u,  we  have  as  in  Art.  308 


and          -- 
du  du 


It  follows  *  that 

2^  =  b  +  2cx 
du2 

=  b  +  2cy 


If  next  we  introduce  two  new  variables  defined  by 

p  =  x  +  y     and     q  =  x  -  y, 
we  have 


•        =  X  -  Y  =  bq  +  cpq  +  ±qd(3P2+  q2)  +  $  epq(p2  +  q2). 
It  is  seen  at  once  that 

A-ittLriftiirt, 

du2      du  du 


...tf+2  , 

fdu?du      q*\du)  du  P)du 

*  See  Cayley,  loc.  tit.,  p.  337. 


366  THEORY    OF   ELLIPTIC    FUNCTIONS. 

The  integral  of  this  expression  is 


where  C  is  the  constant  of  integration. 

Writing  for  q,-£»p  their  values,  we  see  that  the  general  integral  of  (I)  is 


(II)  f-^S I_J_  I  =  c  +  d(x  +  y)  +  e(x  +  ?/)2. 

Cayley  (Elliptic  Functions,  p.  338)  gives  several  interesting  forms  of  this 
algebraic  integral  and  of  the  addition-theorem. 

ART.  310.     The  formula  (II)  above  suggests  at  once  a  form  for  the  inte 
gral  of  the  corresponding  differential  equation  in  the  Weierstrassian  theory . 

Write  (a,  6,  c,  d,  e)  =  ( -  g3,  -  g2,  0,  4,  0) 

and  consider  the  integral 

(10  ds      —  +  dt      —  =  o. 

V4  s3  -  g2s  -  g3      V4  t3  -  g2t  -  g3 
The  algebraic  integral  is  seen  at  once  to  be 


s  —  t 
Writing 


,  ds  ,  dt 

du  =  —  =^  =  .      dv  = 


V4  s3-  g2s  -  g3  V4  t3-  g2t  -  g3 

the  transcendental  integral  is 

(T)  u  +  v  =  c, 

where  s  =  <@u,     t  =  pv  =  $>(c  —  u)=  @(u  —  c). 

When  these  values  are  substituted  in  the  algebraic  integral,  it  becomes 

(A)  Wu-v'(c-UW_  4fw  _  4j,(e  -  *)-  C. 

\_  @u  —  %>(c  —  u)  J 

From  (A)  it  follows  (Art.  300)  that  C  =  4  g?(c),  and  from  (T)  we  have 


or 

2  (@u  —  @v)2 


THE   ADDITION-THEOREMS  367 

ART.  311.     Equate  to  zero  the  determinant* 
1,  pu,  p'u 

=  pfw(pv  —  pu)  +  pfv(pu  —  pw)  4-  p'u(pw  —  pv)  =  0. 


1,  pv,  p'v 
1,  pw,  p'w 

Squaring  we  have 

(p'w)2(pu  —  pv)2—  {p'v(pu  —  pw)-  p'u(pv  -  pw)  }2=  0, 
or 
(pu  —  pv)2[4p3w  —  g2pw  -  g3]  -  [p'vpu  -  p'upv  -  pw(p'v  —  p'u)]2=  0, 

an  equation  which  is  satisfied  for  w  =  u,  v  and  also  (Art.  301)  for  —w  = 
u  4-  v;  that  is  for  pw  =  pu,  pv,  p(u  4-  v). 
The  equation 

(pu  -  pv)2{s  -  pu}{s  -  pv}  {s  -  p(u  +  v)}=  0 

has  the  same  zeros,  viz.,  s  =  pu,  pv,  p(u  4-  v);  and  since  the  coefficients  of 
(pw)3  and  s3  are  the  same  in  both  equations,  the  two  equations,  since 
they  can  differ  only  by  a  multiplicative  constant  (Art.  83),  must  have  all 
their  coefficients  the  same. 

The  coefficients  of  (pw)2  and  s2  give  immediately 

-  (p'u  -  p'v)2=  4  (pu  -  pv)2\-pu  -  pv  -  p(u  +  v) }, 
or 


^  (  pu  —  pv 
ART.  312.     In  Art.  193  we  derived  the  formulas 

u(+l)=-3K  or     sn(- 3/0=1, 

--  —  3  K  —  iK'     or     sn  ( -  3  K  —  iK')  =  - » 

n/ 

u( oo,  GC  )  =  —  iK'  or     sn  ( —  iKf)  =  oc  , 

u(Q,  1)    =  0  or     sn(0)=  0. 

Ti(—  1)   =  —  K  or     sn(—K)  =  —lj 

1\            -..       .„,                          „       .r.,N          1 
- )  =  —  K  —  iK        or     sn  (—  K  —  iK)  = , 

kj  k 

u(<*j,  —  (x)  =  —  2K  —  iK'     or     sn(—  2  K  —  iK')  =  oo, 
u(Q,-l)=-2K  or     sn(-2K)=Q. 

By  means  of  these  formulas  and  the  addition-theorems  we  may  verify 
the  formulas  IX-XV  of  Chapter  XI. 

*  See  Daniels,  Am.  Journ.  of  Math.,  Vol.  VI,  p.  269. 


368  THEORY   OF   ELLIPTIC   FUNCTIONS. 

ART.  313.     Duplication. — In  the  addition-theorems  above  if  we  write 
i)  =  u,  we  deduce  the  following  formulas: 

2  snucnudnu 

D 
cn2u  —  sn2u  dn2u 

~D~ 
dn2u  —  k2sn2ucn2u 


cn2u 
dn2u 


D 
Writing  sn  u  =  s,  en  u  =  c,  dn  u  =  d,  we  have 


1  +  dn  2  u  = 


D  '  D 

2d2 


D 
ART.  314.     Dimidiation.  —  From  the  above  formulas  we  deduce  at  once 


or 


cn2u  = 


1  +  dn2u  2         I  +  dnu 

dn2u  +  cn2u  k'2  +  dn2u  +  k2cn2u 


1  +  dn2u  I  +  dn2u 


Changing  uto^u  we  have  formulas*  for  the  determination  of  sn(%K), 
sn(%  iK'),  sn  (%  K),  etc.;  for  example 


£-\A 

2        Vl 


•  =  -%/=r=^f  • 

+  dnK 

ij?  =     /dniK'+cnJK'        /-  ikl  -  U        /I  +  k 
1  2        V      1  +  drciK'          V     I -ikl         \      k 

[Table  of  Formulas,  No.  XVII.] 


In  a  similar  manner  we  have 


_ Vk  +  ik 


where  we  have  written 


k  =  Vl  +  k'  Vl  -  k',        1  =  Vk  +  iA 
and  noted  that 


Vk  -  ik'  +  Vk  +  ik'  =  Vl  -  k'  +  Vl 
*  See  Table  of  Formulas,  No.  XVII. 


THE    ADDITION-THEOREMS. 


369 


ART.  315.     To  determine  the  value  of  the  ^function  for  the  quarter- 
periods,  we  note  that 


c^  , 


k  = 


We  have  for  example 


SK 


(t) 


€1-63 


-  «s)d 


—  63)  (e  i—  e2); 


en 


=  -  2i(el-e3)ikk'(k-ik') 


or 


a  formula  which  is   incorrectly  derived   and   given  by  Halphen,  Fonct. 
Ellipt.,  t.  I,  p.  54. 
ART.  316.     We  also  find  that 


to)  = 


sn2(v  +  K) 
+  (gi—  es)dn2v  =  e  + 


[v  =  u  Vei  -  e3] 


It  foUows,  if  we  write  5(0  =  4(«  -  ed(t  -  e2)(t  -  e3),  that 


\ 

U  +0})= 


and  similarly 


4   jpM  -  €1 

1     S'(e2) 

-    -  L^Z~: 

4  g?w  -  e2 


4  pi*  -  e3 


370 


THEORY   OF  ELLIPTIC   FUNCTIONS. 


If   further  we  let  P\(u}  =  $>u  —  e\     (X  =  1,  2,  3),  we   may  derive  at 
once  the  formulas  * 


P2(u 


4 


=  («!-  e2) 


Pi(u) 


Ps(« 


=  («2 —  e3) 


Pi(u) 


«  aw 


EXAMPLES 


1.  Show  that 

2.  Show  that 

3.  Prove  that 


+  v)  = 


snv  cnu  dnu  —  snu  cnv  dnv 

1  12  k2cnu  cnv 

cn(u  +  v)       cn(u  -  v)       dn2u  dn2v  -  k'2 

cn(u  +  v)       cn(u  —  v)  _  2  snu  cnu  dnv 
sn(u  +  v)      sn(u  —  v)          sn2u  —  sn2v 


,          .       1   d  i      1  +  k  snu  snv 

4.    Prove  that      sn(u  +  v)  —  sn(u  —  v)= log 

k  du        1  —  k  snu  snv 


5.   Prove  that 


tan  am 


u  +  v  _  snu  dnv  +  snv  dnu 
2  cnu  +  cnv 


6.  Verify  the  formulas  given  in  the  Table  of  Formulas,  No.  LXIII. 

7.  Derive  the  addition-theorem  for  the  g?-function  from  that  of  the  sn-function. 

8.  Show  that 

®2(0)  H(i?  -  u)  H(v  +  u) 


v  —  sn*u 


*  See  also  Art.  327. 


THE   ADDITION-THEOREMS. 


371 


9.  If  am  a  =  a,  am  b  =  /?,  am(a  +  b)  =  cr,  show  that 

(1)  sin  a  sin  /?  A  a  +  cos  o  =  cos  a  cos  /?, 

(2)  cos  /?  cos  <T  +  Act  sin  /?  sin  a  =  cos  a, 

(3)  ACT  +  A-2  sin  a  sin  /?  cos  <r  =  Aa  A/?. 

10.  Show  that  the  algebraic  integral  of 


where 


X  =  a^  +  4  a^3  +  6  a2x*  +  4  a.^  +  a4, 
y  =  a0y4  +  4  a^3  +  6  a,?/2  4-  4  a3y  +  a4, 
may  be  expressed  in  the  form  of  the  symmetric  determinant 


X   ~\~  It 

o, 

^                  2' 

xy 

1, 

a0>              alf 

a2-  2c 

x+y 

alt              a2+  c, 

«3 

2 

vy, 

a2-  2  c,     a3, 

«4 

(Lagrange.) 


=  0, 


where  c  is  an  arbitrary  constant  (Richelot,  Crelle,  Bd.  44,  p.  277;  Stieltjes,  Bull, 
des  Sciences  Math.,  t.  XII,  pp.  222-227). 


CHAPTER  XVII 
THE    SIGMA-FUNCTIONS 

ARTICLE  317.  In  Chapter  XIV  we  derived  the  function  ou  from  a  certain 
theta-function  and  we  then  proceeded  to  the  other  sigma-functions.  In 
Chapter  XV  the  function  ou  was  denned  through  an  infinite  product 
which  followed  from  the  definition  of  the  ^-function  and  the  character 
istic  properties  of  the  sigma-function  were  thus  established. 

We  shall  now  prescribe  these  characteristic  properties  of  the  sigma- 
functions  and  derive  therefrom  directly  the  functions  themselves.* 

In  Art.  298  it  was  shown  that 


We  write  v  =  &,  where  2  5  =  2  pa)  +  2qajf.  The  quantities  p  and  q  are 
integers,  and  here  one  of  them  at  least  is  taken  odd,  so  that  &  is  different 
from  a  period. 

Since  a>  is  a  half  period,  we  may  write 

pa>  =  ei         (i  -  i,  2,  3). 
The  formula  above  becomes 


In  Art.  290  we  derived  the  formula 

a(u  +  2  6>)  =  = 

where  2  rj  =  2  py  +  2  qy',  and  the  negative  or  positive  sign  was  to  be  taken 
according  as  oco  was  different  from  or  equal  to  zero. 

In  the  present  case  we  must  therefore  take  the  negative  sign;  and  if 
u  —  a>  is  written  for  u,  it  follows  that 

a(u  +  a>)  =  —  e2JlU  o(u  —  a>). 
We  consequently  have 


o^u  azw         \ 

*  Hermite  (p.  753  of  Serret's  Calculus,  2d  volume,  1900)  writes:  "  Nothing  is  more 
important  nor  more  worthy  of  interest  than  a  careful  study  of  a  process  by  which, 
starting  with  notions  previously  acquired,  one  comes  to  the  knowledge  of  a  new  func 
tion  which  becomes  the  origin  of  a  new  order  of  analytic  notions." 

372 


THE   SIGMA-FUNCTIONS.  373 

If  a\u  is  defined  through  the  equation 


aoj 
we  have 

(\  2 
au/ 

The  quantities  77  and  if  are  defined  as  in  Art.  259.  As  there  are  only  three 
incongruent  half-periods,  we  have  the  three  new  functions 

ffitt          (/  =  1,  2,  3). 

When  u  =  0  it  is  seen  that  a/u  =  1.  We  defined  in  Art.  272  the  function 
au  through  the  relation 

_  _  d2  log  au d_  a'u  _  aua"u  —  (a'u}2 

d2u  du  au  (au)2 

If  then  we  require  that  the  sigma-functions  be  one-valued,  analytic  func 
tions  which  have  the  character  of  integral  transcendental  functions,  it  is 
seen  that  $>u,  pu  —  e\  may  be  expressed  through  the  quotient  of  such 
functions  (Art.  262). 

ART.  318.  By  means  of  Laurent's  Theorem  we  may  express  at  once  the 
function  au  through  a  Fourier  Series  as  follows: 

If  f(t)  is  a  one-valued,  finite  and  continuous  function  within  and  on  the 
boundaries  of  a  ring  inclosed  between  two  circles,  it  may  be  developed  in 
a  series  *  consisting  of  an  infinite  number  of  positive  and  negative  terms 
in  the  form 


f(t)=   J    ck(t  —  a)k     (ck  constant). 

fc— 00 

We  shall  next  assume  that  the  interior  circle  is  arbi 
trarily  small,  so  that  the  above  series  is  convergent 
for  the  entire  larger  circle  with  the  exception  of  the 
point  a. 

Let  F(u)  be  a  function  defined  for  the  whole  or  a  Yis  72 

part  of  the  w-plane  and  suppose  that  this  function  is 
one-valued,  finite,  continuous  and  simply  periodic  having  the  period  p,  say. 

We  then  have 


2xt 

p 


If  we  write  (cf  .  Art.  67)  t  =  e  p    ,  or  u  =  -—.  log  t,  we  have 


*  Osgood,  loc.  cit.,  p.  295. 


374  THEOEY    OF   ELLIPTIC   FUNCTIONS. 

The  function  f(t)  is  one-valued,  for  if  a  definite  value  I  is  given  to  t,  then 
u  =  ^—  .log  I  +  kp     (k  an  integer). 

But  for  all  such  values  the  function  F(u)  retains  the  same  value,  since 
p  is  its  period.     It  follows  that  F(u)  is  one-valued. 
Further,  if  t  describes  a  circle,  so  that  t  =  re^,  then  is 


or  u  =  b  +  m<j)     (b  and  m  constants); 

and  consequently  u  describes  a  straight  line  [Art.  60]. 

From  the  relation  u  —  —  =  log  t  it  is  seen  that  for  t  =  0  and  also  for 

P 
t  =  GO  we  have  u  =  oo  ;  and  since  u  =  oo  is  an  essential  singularity  of  F(u)t 

it  follows  that  t  =  0  and  t  =  oo  are  singularities  of  f(t)  . 

Since  zero  is  a  singular  point  of  f(t)  ,  we  have  from  above  the  expansion 


fc— 00 

and  therefore  A=+X 


ART.  319.     We  write 


and  we  shall  so  determine  the  constants  A,  B,  C  that  <f>(u)  has  the  period 
2  w.     This  function  ^(w)  is  one-valued,  finite  and  continuous  for  the  finite 
portion  of  the  it-plane. 
From  the  formula 


we  have,  since 

ou, 

the  formula 


It  follows  that 

2  TJ(U  +  to)  +  2  Bco  +  4  CUM  +  4  Cw2 =  (2  &  +  l)7rt, 
where  fr  is  an  integer;  and  consequently 

2  T}  +  4  Cw  -  0,     or     C  =  -  -  2  ; 

2  w 

and  2  ^  +  2  Bco  +  4  Co>2=  (2  k  +  I)TTC; 

or,  if  k  =  0,  „  =  7ii_ 

The  remaining  constant  A  being  arbitrary,  may  be  taken  equal  to  zero. 


THE   SIG  MA-FUNCTIONS.  375 

We  then  have  ,        -i 


--  -  u-  H  --  u 

=  one 


We  further  write  u  =  2  wv  and  put 

0(u)  =•*•(!,'). 
Since  <j>(u  +  2aj)  =  <j>(u),  it  follows  that 

<$2a)(v  +  1)]  =  0(2o>v), 
or 

Vr(t>  +  l)  =  tM, 

and  consequently  from  the  last  Article 

0  (P=   1), 


a  series  which  is  uniformly  convergent  within  the  finite  portion  of  the 
r-plane. 

To  determine  the  coefficients  Ck,  we  note  that 


o(u  +  2o/)=-  e  , 

and  consequently 

^-  (M+2o/) 


or 

w'       '^w'2  at' 

2jj'(ti+a/)-2ijtt  ---  jj+st  — 


Since  TJOJ'  —  wy'  =  —,  it  follows  that 


Writing  —  =  T,  we  have 
to  . 

<56(2  o>y  +  2  a/)  =  -  e~ 
or 

^r(v  +  T)=  -  r 

Since 

/:=  +  x 


we  therefore  have 

fc=+ 

V 


or 


376  THEOEY   OF   ELLIPTIC   FUNCTIONS. 


If  the  coefficients  of  e2*™  on  either  side  of   this  equation  are  equated, 
we  have  -  CA  =  CA-I  e2^-1^, 

which  is  a  transcendental  equation  of  differences. 
In  the  formula  ^^  _  (7A_1g2(A-i)«t+B»    . 

change  ^  to  A  +  1  and  write  log  CA  =  #A-     We  then  have 


Suppose  that  ^(A)=  A0+  A!  A  +  A2A2  and  consequently  that 
^(/l  +  1)-  %(*)  =  AI+  2A2A  +  A2=  2^rir  +  ?ri. 

It  follows  that 

A2  =  Ttt'r     and     AI=  7ri(l  —  r). 

As  A0  remains  arbitrary,  we  choose  it  equal  to  zero.     These  values  sub 
stituted  in  y(A)  give 


Let  us  further  write  Bx  -  j(A)  =  EL     We  then  have 
#A+1=  Bm-  /(A  +  1)=  B 

-Bi 

We  note  that 


Further,  since  BA  =  ^(/)  + 

or 

we  have 

Writing  eE«=  C,  it  follows  that  CA  =  Ce"1'^-0^^^,  and  consequently 


Further,  since 

•^r(v)= 

it  is  seen  that 

CTI^  =  a(2ow)= 


?rtt 
4 


/2fc-l 

2 


THE   SIGMA-FUNCTIONS.  377 

_ru 

Letting  —  e    4  C  =  c  and  substituting  k  +  1  for  k,  we  have 


We  note  that  era  is  an  odd  function  and  we  shall  assume  that  the  constant 
c  is  such  that  the  coefficient  in  the  first  term  in  the  expansion  of  au  is  unity, 
that  is, 

au  =  1  •  u  +   •  •  •  . 

The  sigma-function  is  thus  completely  determined. 

ART.  320.     If  we  write  £  =  w,  M",  co',  we  have  directly  from  Art.  317 
the  formulas 


OOJ  OOJ 


where  w"  =  w  +  a;'  and  ^r/  =  y  +  yf. 

The  argument  2  cj(v  +  })  corresponds   to   the  argument  u  +  a).     We 
may  consequently  write 


so  that 


or 

,      i)ta      rt    k  =  +oo 

C/>^)l7£tJT7'"-r  ~7T"  T  TT 
c  22 


If  we  write 


then  is 


and  similarly  k=+x 

a2u  =  t32e2rjujl''  2)  c 
*--* 

fc=+oo 


fc=-M 


378  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  with  Weierstrass  we  write 

e*n*  =  h     and     ealv  =  z, 
we  have 


£=-00 


Tit    —  —  7T  -ZT- 


Using  the  notation  of  Jacobi:  h  =  e    "  =  e     K  =  q,  and  writing  with  him 


=  2  g*  sin  nv  —  2  5^  sin  3  xv  +  2  g¥  sin  5  TTV  —  •   •   •  , 

fc=+oo     (2fe+l)2 

-  2)  9    4    ^2*+1 

A;=-oo 
=  2^*  COS  KV  +  2  g*  COS  3  7TV  +  2  g^  COS  5  7TV  +  •    •    • 


=  1  +  2  5  cos  2  TTV  +  2  g4  cos  4  TTV  +  2  q9  cos  6  ^v+  •   •  •  , 


=  1  —  2  ^  cos  2  TTU  +  2  54  cos  4  TTV  —  2  <?9  cos  6  TTV  +  •   •   •  , 
we  have 


ART.  321.     By  differentiating  both  sides  of  the  formula  above  for  era 
and  then  writing  u  =  0,  it  is  seen  that 


du 
3=^- 

p     «V(0) 


THE    SIGMA-FTXCTIOXS.  379 

Further  since  ^i(0)=  1  =  tfo(0)=  ^s(O),  it  follows  that 


#2(0)  03  (0)  00(0) 

In  Art.  340  it  is  shown  that 

$!'(0)  =  2  77/Z*  JJ  (1   -  ft2")3. 
n  =  l 

When  this  value  is  substituted  in  the  formula  above  for  mi,  we  have 

n  =  x 


In  a  similar  manner  if  we  write  for  4#i,  /?2,  /?3,  their  values  we  have 


cos  2  «r 


<72u=^-2jj— 

n=  1 
G3li  =  e2v**  JJ  1^ 

Take  the  logarithmic  derivatives  of  oiu,  o2u,  o^u  and  equate  the  coeffi 
cients  of  u  on  either  side  of  the  resulting  expressions.     We  then  have 


Since  ei  +  e2  +  e-3  =  0,  it  follows  from  Art,  286  that 


3  h       (i  + 


We  note  that  au  is  an  odd  function,  while  aiu,  a2u  and  o%u  are  even  func 
tions. 

The  zeros  of  these  four  functions  are  given  in  the  Table  of  Formulas, 
No.  XXXI. 


380  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

ART.  322.     If  the  formulas 


/        9/V 

<@u  —  e\  =  (  — L^ )  j     <@u  —  e2  =  [  -^^ )  i     <@u  —  63  =  ( — ^—  J 
\  au  /  \ou  /  \au  / 

are  multiplied  together,  we  have  in  virtue  of  the  equation 


the  formula 


To  determine  the  sign  to  be  used  before  the  root,  write  u  =  0  and  it  is  seen 
that  the  negative  sign  must  be  employed.     We  thus  have 


(1) 


°*u 


O6U 

In  Art.  302  it  was  seen  that 


It  follows  from  (1)  that 

cr  (2  u)  =  2  au  GIU  a2u 


ART.  323.     We  may  next  note  how  the  sigma-f  unctions  behave  when 
the  argument  u  is  increased  by  a  period. 
Since 


OOJ  OOJ 

it  follows  that 


OU) 


<7W  (70J 

or 


and  similarly  * 


Formulas  for  a\(u  +  2w"),  etc.,  are  found  in  the  Table  of  Formulas, 
No.  XXVI. 

*  See  Schwarz,  loc.  cit.,  p.  22. 


THE   SIGMA-FUNCTIONS.  381 

ART.  324.     Let  X,  //,  v  represent  in  any  order  the  integers  1,  2,  3;  then 
by  Art.  262  we  have 


(\  2 
—  j  =  $u  — 
ou  / 


ou 


no  two  of  the  quantities  X,  a,  v  being  supposed  equal. 

By  eliminating  <@u  from  the  second  and  the  third  of  these  formulas  we 
have 

I  j  (  )     =          W/l          &v)) 

or 


_  Cy)  (j2M  =  0. 

It  is  also  seen  that 

(e2—  e3)oi2u  +(e3—  6i)a22u  +(e\—  e2)cr32u  =  0. 


DIFFERENTIAL  EQUATIONS  WHICH  ARE  SATISFIED  BY  SIGMA-QUOTTENTS. 
ART.  325.     If  the  formula 


is  dirTerentiated,  and  for  <@'u  its  value  in  terms  of  the  sigma-functions  is 
substituted,  it  follows  that 


ou      ou  ou    ou    ou 

or 

du  ou  ou    ou 

If  we  differentiate  the  equation 


ovii         &u  -  e 
it  follows  that 

2  a^u  (L_  a^  =  (g?M  -  ev)  -  (pu  —  eu}    , 
ovu  du  avu  (pu  —  ev}2 


(o_vU\ 
\ou) 


or 

1 tL-  =  —  (e^—  ev)  ~^— 

du  ouu  ouu 


382  THEORY   OF   ELLIPTIC   FUNCTIONS. 

From 


it  follows  that 


2    j  avii 


d  /  ou\  _  _         <@fu        _      ou    ou    ou 
du\oxu)          (<@u  —  ex)2  /<?A*A4 

(ou) 


or 

d    ou  OfU  ovu 

du  OxU       oxU  OxU 
Since  the  equation 

o2u  —  o^u  +  (e^—  ev)o2u  —  0 
may  be  written 

o.2u  ,  v  o2u 


and  further  since 
we  have 


OfU 


Cj  1       ( 

au  oxu/       \o 

(~r  — )  =    1  ~  fc~  ^A)[  — 
d^    (T^/  |_  \GxU/ 

In  the  same  way  it  may  be  shown  that 
I  -I        \2     T~  2~ir~ 

\d       IL~)   =  ^2~      C' 

2"4          el 
^T        A       "J 


and 


It  follows  that 

OU  1  OM  1  <T.,W  1 


(e,-^)(ey-^)   ou 
are  all  particular  solutions  of  the  differential  equation  * 

(A)  (^ Y  =  [1  -  (e,-  ex)?2]  [1  -  (e, - 


ART.  326.     If  we  write  v  =  2,  /£=!,  >l  =  3  and 

-. 
s    = 

the  differential  equation  (A)  becomes 


*  See  Schwarz,  loc.  cit.,  Art.  25;  or  Daniels,  Am.  Journ.  Math.,  Vol.  VI,  p.  180  and 
Vol.  VII,  p.  89. 


THE   SIGMA-FUNCTIONS. 


383 


Further  write 


and 


(ei-e3)c2=*2,     or     ^ei- 


We  then  have 


If 


w=  r 

J0  V7(l  - 


-  k2x2) 


then  is  x  =  sin  am  UOT  x  =  sn(u,  k). 
We  therefore  have 


(1) 

Further,  since 

and  as 

we  have 

(2) 

and  similarly 

(3) 

ART.  327.     If  we  write  * 


=  0, 


-  e3 


— 
011 


—  e 


we  have  at  once 
and 


*  See  Enneper,  Elliptische  Funktionen,  p.  160;  or  Tannery  et  Molk,  Fonct.  EUipt., 
t.  II,  Chap.  IV. 


384  THEORY   OF   ELLIPTIC   FUNCTIONS. 

It  is  also  evident  that 

3  <@u  =  £iQ2(u)  +  £n(?(u)  +  £vQ2(u), 
2  a)))  =  £),Q(U),     )  where  we  write  without  regard  to  order 

V  f  ft      f 

)  (OX)  (ou)  (Ov  lor  oj.  d)     d) . 


£fiV'(u)  =  -  (eft-  ev) 
ART.  328.     Through  the  equations  * 


CD 


an 


— ,    v ,«,«,- 32, 

<7W  GU 


the  values  of  the  three  quantities  \/$u  —  e^  are  denned  as  one- valued  func 
tions  of  u. 

If  we  give  to  u  the  values  w,  01",  a>',  it  is  seen  that 


(2) 


GOJ 


/ —  <7iW  6'w    Gd) 

GO)"  0(DO(i)" 


Ve3-€,= 


GI(I)'       e' 


GO)  0(0  OO) 


e\—  63  = 


63—  e2  = 


GO) 


0(1) 


Through  these  formulas  the  six  quantities  on  the  left-hand  side  are  uniquely 
determined. 
We  note  that 


On  the  other  hand 


i (*)  — 


if 


'€3-  e2 


fe2-  e3 


iaj 


(see  Art.  288). 


Hence  among  the  six  quantities  above  there  exist  the  relations 

v  63—  62=  —i  ve2—  63,  V 63—  e\=  —  i \/e\—  63,  *ve2—  e\=  —i 


or 


—  e2=  \e2—  e^e^—  e\=  \e\—  e3,e2—  ei  =     e\—  e2. 

We  have  thus  reduced  the  six  roots  without  any  ambiguity  to  the  three 
roots  \/e\  —  e2,  \^e\—  63,  ^/e2—  63,  which  three  roots  are  real  and  posi 
tive  if  the  discriminant  of  4  s3  —  g2s  —  g3  =  0  is  positive. 

*  Schwarz,  loc.  cit.,  Art.  21. 


THE   SIGMA-FU3TCTIONS.  3S5 

Remark.  For  the  sake  of  a  greater  symmetry  some  recent  writers  on 
this  theory  have  written  w\,  oj2,  MS  for  the  quantities  which  at  the  outset 
with  Weierstrass  we  denoted  by  w,  a>" ',  w'.  When  such  formulas  that 
result  are  compared  with  those  given  by  Weierstrass,  much  confusion,  in 
particular  with  regard  to  sign,  arises;  for  example  with  these  writers 


^€3  —  e2  =  i  V^2  —  €3,  v  €3  —  ei=  —  i  >ve\  —  €3,  V  'e2  —  e\  =  i  \/e\  —  e2. 

The  explanation  they  give  to  —  a>2  is  not  entirely  satisfactory,  especially  if 
these  quantities  are  defined  on  the  Riemann  Surface  with  reference  to  K 
and  iK'. 

ART.  329.     From  the  equations  (2)  above  it  follows  that 


f^  e'  „ 

(A)  aw  =  ,,  ,    war'  = 


Vie**""" 


=  4  .  4/ 

Ve2-  63  vei—  63 

We  note  here  (see  also  Art.  345)  that  the  quantities 


4/  -  -     4/  - 

Ve2  —  63  Vei  —  e2 
(where  i  =  e^). 


can  take  only  such  values  whose  squares 


e2  -  63,        ei  -  e3,        61-62 

are  uniquely  determined  through  the  equations  (2)  of  Art.  328.  Hence  each 
of  the  fourth  roots  may  take  two  and  not  four  values;  but  as  soon  as  the 
value  of  any  one  of  these  quantities  is  known,  the  values  of  the  two  others 
are  uniquely  determined  through  the  formulas  (A). 

If  in  the  formula 

e~*uo(u)  +  u) 

G\U  =  -  *  -  !  -  L% 

aw 

we  put  u  =  —  J  w,  we  have 


It  follows  that  we  may  write  formulas  (A)  in  the  form 

^, 

•(¥) 

"\/ei—  63  Vei—  ^2=  —  j—  ?'     Ve2—  e3  Vei—  e2  = 

/oA 

a3(") 

•                                     „-,     .......v^. 

which  expressions  may  be  used  to  determine  the  products  of  any  two  of 
the  three  fourth  roots. 


386  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  330.  We  may  next  derive  a  table  of  the  four  sigma-functions  when 
the  argument  is  increased  or  diminished  by  a  quarter-period.  It  is  assumed 
that  the  definite  values  derived  above  are  given  to  the  square  and  fourth 
roots  that  appear. 

Take,  for  example,  the  formula 


e~1iu(j(co  +  u)  _  6^0(0}  —  u) 
o\ii  =  -- 

Git)  OCO 


We  have  at  once 

o(u  ±  co)  =  ± 
Further,  since 


it  follows  that 

-e2 


The  formulas  given  in  the  Table  of  Formulas  No.  XXXIV  should  be 
verified. 

ART.  331.     It  is  seen  that 

a(u  +  2co)   =  _   cru 

o3(u  +  2  co)  GSU 

and  consequently 

<ju  +  4a>          ou 


(73(u  -\-4co)       o%u 

Also,  since  a^u  +  2  w'^   =  —>  it  follows  that  4  CD  and  2  a/  are  periods  of 
<73(w  +  2  to') 


— •    A  closer  investigation  shows  that  4  co  and  2  co'  are  a  primitive  pair  of 

periods  of  this  function;  for  in  the  period-parallelogram  with  the  sides  4  co 
and  2  w'  the  function  <j3w  becomes  zero  only  on  the  points  co  and  2  co  +  co', 

being  zero  of  the  first  order.     Hence  —  becomes  infinite  of  the  first  order 

<73lt 

on  these  points.  Since  only  two  infinities  lie  within  the  period-parallelo 
gram  with  the  sides  4  co  and  2  co' ,  and  since  the  smallest  number  of  infin 
ities  within  a  primitive  period-parallelogram  is  two,  it  follows  that  4  co,  2  co' 

form  a  pair  of  primitive  periods  of 

ART.  332.     It  follows  at  once  from  the  formulas  above  that 

a(u  -f  co)    =          1  o\u 

This  may  also  be  seen  from  the  formula  of  Art.  326 


THE    SIGMA-FUNCTIOXS.  387 


Since  K  =\/e\  —  e  3  •  a>  we  have 


1 


V  '  e\— 
For  u  —  0,  it  follows  that 


1 


ei  —  €3 

and  further  that  all  values  of  u  which  satisfy  the  equation 

OIL  1 


<73u          ei—  e3 
are  contained  in  the  form 

a)  +  Ipw  H-  2qw', 

where  p  and  q  are  integers  positive  or  negative,  including  zero. 
We  might  define  K  more  generally  through  the  equation 


K  =      ei—  e3(cu  +  4pw 

where  it  is  assumed  that  4  p  +  1  and  2  5  have  no  common  divisor..    The 
quantity  V  fe\—  €3  is  to  have  the  same  value  as  given  in  formulas  (2)  of 
Art.  328  or  the  opposite  value  according  as  q  is  even  or  odd. 
ART.  333.     It  also  follow  from  the  equation 


sn  am 


—  e%  •  u,  k) 


that  —   1         =  —  sin  am  (K,  k), 

Vei—  e3      \  <?i-  e3 

or  sn(K,  A')  =  1     (see  Art.  218). 

The  coamplitude  is  defined  by  Jacobi  (Werke  I,  p.  81)  through  the  formula 
(see  Art.  221) 


—  e3  •  u,  k)  =  am  (K  —Vei-  e3  •  u,  k}, 

—  e3  •  u,  k)  =  am  \Ve\  —  e3  (to  +  4po;  +  2  5^'  —  u),  k]. 


Since  4pa>  is  a  period  for  all  the  sigma-f unctions,  it  may  be  dropped  from 
the  argument  u. 

We  then  have  

K  =v/ei—  e3(u)  +  2qa)r), 

and  

coam  (\/e\  —  e3  •  n,  k)  =  am  [V«i  —  e3  (aj  +  2  qa)r  —  u),  k]. 


388  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

We  may  note  that 

r   /  -  /      ,   r>      /        \   7!      oi(—  u  +  co  +  2,qco'.k) 
cos  am  [V^!-  e3(co  +  2qco'-  u),k\  =  -^  -  -  -       *   /    ' 

<?3(  —  u  +  co  4-  2qco  ,  k) 

_  (jl(u  —  oj  —  2qco',  k) 


a3(u  —  co  —  2qcof,  k) 
Since 


a3(u  —  a)) 

we  have 

r  /  -        T~\        /  - 
coscoam  vei  —  e»  •  u,  k\  =  Ve1  —  e2 
L 

and  since 


o(u  +  2tt>r)   _         ou 


we  have  finally  * 


cos  coamK/e!—  e3  •  w,  A;]=  (—  l^V^—  e2 

<T2^ 

Making  q  =  0,  we  have  the  set  of  formulas  given  in  the  table,  No.  LIV. 
ART.  334.     In  Art.  79  we  wrote  (Cf.  Schwarz,  loc.  cit.,  Art.  33) 

co  =  pco  +  qco',     to'  =  p'to  +  q'co',     to"  =  to  +  a)', 

where  p,  q,  pf,  q'  were    any  integers  such  that  pq'  —  qp'=  1 ;  and  it  was 
seen  that  2  co,  2  co'  and  2  co,  2  co'  formed  equivalent  pairs  of  primitive  periods. 
We  shall  further  write 

TJ  =  pr)  +  qy',     Ij'  =  p'-t)  +  q'rf,     TJ"  =  rt  +  TJ'. 
If  in  the  place  of  the  quantities 

o),a)',  co"=  co  +  to'',        r),  T)',  7)"=  T)  +  rf\ 
we  substitute 

co,  wf,  Z>"=  at  +  £';     Tj,  ij' ,  rj" '=  y  +  T)', 

it  follows  at  once  (Arts.  276,  271)  that  the  invariants  g2,  93  and  the  func 
tions  §m,  ou  remain  unaltered. 
Also  owing  to  the  equation 

(p'u)2=  4[pu  -  pa>][pu  -  v<o"][pu  -  pcof]  =  4[$>u  -  ei][yu  -  e2][pu-e3] 

the  collectivity  of   the  three  quantities  e\,  e2,  e3  remains  unchanged  and 
consequently  also  the  collectivity  of  the  three  functions 

(W\2==pu_eji  (J  =  i,2,3), 

\ou] 

although  the  indices  1,  2,  3  may  be  permuted. 

*  See  Schwarz,  loc.  cit.,  p.  30;  or  Daniels,  Am.  Journ.  Math.,  Vol.  VII,  p.  89. 


THE   SIGMA-FUNCTIONS. 


389 


We  therefore  have  a  set  of  more  general  formulas  if  in  the  preceding 
developments  we  write 


0),        CO 


01,        Op, 

u 

v  =  —  r 
2co 


in  the  place  of 


CO,  CO 

n,  V, 

B\j  69, 

V  =  — 


co 


where  ^,  /*,  v  may  take  in  any  order  the  values  1,  2,  3.     The  corresponding 
changes  must,  of  course,  be  made  in  z  and  h. 

The  following  table  contains  the  values  of  the  indices  X,  p,  v  for  each 
of  the  six  different  cases  which  may  arise  (see  also  Halphen,  loc.  tit.,  1. 1., 
p.  262): 


Residue,  mod.  2 

I 

P 

9 

pf 

q' 

/ 

ft 

y 
3 

1 

0 

0 

1 

1 

2 

II 

1 

0 

1 

1 

1 

3 

2 

III 

1 

1 

0 

1 

2 

1 

3 

IV 

1 

1 

1 

0 

2 

3 

1 

V 

0 

1 

1 

1 

3 

1 

2 

VI 

0 

1 

1 

0 

3 

2 

1 

ADDITION-THEOREMS  FOR  THE  SIGMA-FUNCTIONS. 

ART.  335.  In  a  similar  manner  as  was  done  in  the  case  of  the  theta- 
f unctions  (Arts.  210)  we  may  derive  theorems  for  the  addition  of  the 
sigma-functions.  These  functions  like  the  theta-functions  do  not  have 
algebraic  addition-theorems. 

If  in  the  identical  relation 

=  0 


(tfU  —  $>U2)  (£>M3  —  &Ul)  +  ($>U  — 

we  make  repeated  application  of  the  formula 

a(u  +  v)  a(u  -  v)  ' 


390 

we  have 
(D 


THEORY   OF   ELLIPTIC    FUNCTIONS. 

u3)  o(u2-  u3) 


o(u  +  HI)  a(u  —  HI) 
o(u  +  u2)  o(u  —  u2) 
a(u  +  U3)  a(u  —  u3) 


u2)  a(ui  -  U2)  =  0, 


an  equation  which  is  true  for  all  values  of  the  arbitrary  quantities  u, 

U2,   U3. 

Through  the  equations 


(2)  u  +  ui=  a,        u  —  ui=b, 

u  +  u2=af,       u  —  u2=b', 
u  +  u3=a",      u  —  u3=b", 


u3=  c,       u2-  u3=  d, 
ui=c',      u3—  ui=  d', 
u2=c",     ui—  u2=  d", 


we  may  define  three  systems  of  four  quantities  each 

o,6,c,d;     a',6',c'X;     a",V',c",d", 
among  which  the  following  relations  exist  (cf.  also  Art.  210)  : 


(3) 


a  =i(a   +  b 
b  =i(a"  +  b" 
c  =ia"-b" 


a - 


d=i(a'-b'-c'-df) 
(3')     a2+  b2+  c2  +  d2=  a'2+  b'2+  c'2+  d'2=  a"2  +  b"2+  c"2  +  d"2, 
If  in  equation  (1)  instead  of  the  quantities 

U  +  U\j       U  —  U\j       U2+  U3,       U2—  U3 

we  write  respectively 

[1]  a,  6,  c,  d-     [2]  a  +  5,  b  +  S>,  c,  d', 

[3]  a  -f  %,  b  +  5",  c  -  wf,  d',     [4]  a  +  5",  6  +  a>",  c  +  5',  d  -  5', 

[5]  a  +  5  +  2  £7,  6  +  5,  c  +  &,  d  —  5;    [6]  a  +  5,  6  +  to,  c  +  a>,  d  - 


THE   SIGMA-FUNCTIONS.  391 

we  have  the  following  relations  given  by  Schwarz,  loc.  cit.,  §  38: 

[A.] 

[\\aaabacad  +        aa'  ab'  ac'  ad'  +  aa"  ob"  ac"  ad"  =  0, 

[2}a{iol)acad          +        afr'otfac'ad'          +  a>a"  a,b"  ac"  ad"  =  0, 

[^o&ojbaj  od         +        afr'ap'a^'ad'        +  a  &"  a  fi"  a  j."  ad"  =  0, 

[4]  Offl  ffjb  a^c  a$       —        a^'ap'a^a^'       +  (e/jt-ev)a)a"aib"ac"cfd"  =  0, 
[5]  («j—  «^|B*£*iC*j4+(«^ej)4FXa#^ 
[6]  ajfLobofloid                afr'ab'aic'aid'  +  ( 

From  [A.]  formula  [2]  follow  without  difficulty: 

[B.] 
(1)  aiwo(u  +  v  +  w)a(u  —  v)  =  a(u 


(2)  owa(u  +  v  +  w^a^u  —  v)  =  a(u 

w)av. 


Professor  Schwarz,  loc.  cit.,  p.  50,  gives  eighteen  other  such  formulas. 
Write  in  [A.],  [2]  the  values 

a=0,       6=      0,       c   =  u  +  v,      d   =  u  —  v, 
a'  =  u,      b'  =  —  u,      c'  =  v,  d'  =  v, 

a"  =  v,       b"  =  -  v,       c"  =  u,  d"  =  -  u, 

and  we  have 

[C.] 

a(u  +  v)a(u  —  v)=  o2u  oj?v  —  apu  o2v. 

The  other  eight  formulas  given  in  the  Table  of  Formulas  LXII  should  be 
verified. 

We  note  that  these  formulas  are  the  analogues  of  the  formulas  (D)  of 
Art,  211.  Scheibner  (Crdle,  Bd.  102,  p.  258)  has  derived  the  Weierstras- 
sian  formulas  from  those  of  Jacobi.  A  method  by  which  the  formulas  of 
both  Jacobi  and  Weierstrass  may  be  derived  is  given  by  Kronecker  (Crelle, 
Bd.  102,  p.  260);  see  also  Briot  et  Bouquet,  Traite  des  fonctions  elliptiques, 
pp.  485  et  seq. 

EXPANSION  OF  THE  SIGMA-FUNCTIONS  IN  POWERS  OF  THE  ARGUMENT. 
ART.  336.     In  Art.  281  we  saw  that 


23.3-5.  7 

and  in  Art.  279  we  saw  that  the  coefficients  of  u  were  rational  functions  of 

g2  and  ^3. 


392 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


We  may  determine  these  coefficients  as  follows :  * 
If  the  equation 

/d@u\2 

\du) 

be  differentiated  respectively  with  respect  to  u,  g2}  g$,  we  have 

2ajerP 


(a) 


=  [12  (»*)*-  <, 

2  g2 


)2        _L 


We  also  have 


du 


dpu 


du 


du 


dgw 
du 


1       1 


We  further  note  that 


(2) 


du 


„ 

du 


dpu 


-3pu  +  — 


6    2w 


+  ±l^u 
"1      ? 


If  the  equation  (2)  is  integrated  with  respect  to  u,  it  becomes 

-»a»+-igri 


or 


92' 


a3  log  on 


(3)^2 

Noting  that 


a3  lo 


aw  dw 

,  a  log  aw  a3  loj 


a2  lo 


it  is  seen  that  the  constant  of  integration  that  would  appear  in  (3)  is  zero. 

a2 

Since  — -  log  ou  =  —  @u,  we  have  from  (a) 


*  See  Weierstrass,  Zwr  Theorie  der  elliptischen  Functionen,  Berl.  Monatsb.,  1882, 
pp.  443-451;  TFerfce,  Bd.  II,  p.  245,  and  also  Forsyth,  Quarterly  Journ.,  Vol.  XXII, 
pp.  1  et  seq.;  Hermite,  Crelle,  Bd.  85,  p.  248;  Meyer,  Crelle,  Bd.  56,  p.  321;  Enneper, 
Ellipt.  Funct.,  p.  166. 


THE   SIGMA-F  UNCTIONS.  393 

and  observing  the  identity 

i  a2  /dlo<rau\2=  a  log  <m  a3  log  au  ,  /a2  log  ^iA2 

2du*\     9"     /  dw  aw3       '  \     du2     )  ' 

it  is  seen  that  the  equation  (3)  may  be  written 

2  a3  log  mi   ,    1Q      a3'log  au  _  3  62  /6  log  <m\2  .   3  64  log  mi   ,   1 

92  ***  =        ~        +          ~  f  ^ 


This  equation  when  integrated  twice  with  regard  to  u  becomes 

2  6  log  au  .    -,  Q      3  log  <m      3  /a  log<7jA2      3  a2  log  <m   .1       2 

^     -  ^  -  +   18^3  -  f  -  =  o    -  ^  -       +  o  -  n  -    +  o^2W     ; 

a^3  dg2  2\      du      /       2       du2  8 

or 

2  al^u     lg    aji^  _  |  ^  a^,  +  i     2 
a^3  agr2        2  a^  a^2     8 

the  constant  of  integration  being  zero. 
It  follows  finally,  since 

a  log  au  =  J_  dau 
a.r  au    dx 

that 


C/2 


Using  this  as  a  recursion  formula  Professor  Schwarz  (loc.  rit.,  p.  7)  has 
calculated  the  terms  of  au,  up  to  the  35th  power  of  u. 
If  with  Halphen  *  we  write 


'  -    o!    '  *7!    '  "(2n+l)! 

we  have 

7      _   -j  9         afon- 1  .2       2  a6n-  1          (2  U  - 

On—    14  y3       I  r  —  J/2     ~ 


To  simplify  the  computation  write 
and  consequently 


02=  2i2,       03=  ~^3, 
o 


n-l"[         (2  71 
xJ  -- 

/?3  j 


2  71  -   1)  (n  -   1) 
xJ  o  - 

a/?3  6 


*  Halphen,  loc.  cit.,  t.  I,  p.  300. 


394  THEORY   OF   ELLIPTIC   FUNCTIONS. 

It  follows  from  (1)  that  62=  -  h2,  63=  -  4ft3;  and  from  (5)  we  have 


=22-32 

23^32  +  107 


Expansions  for  sn  u,  en  u  and  dn  u  were  given  in  Art.  226.  These  functions 
may  be  expressed  as  quotients  of  theta-functions.  We  have  not,  however, 
expressed  the  theta-functions  in  powers  of  u.  As  we  have  already  given 
the  expansions  of  gw,  £u,  etc.,  in  powers  of  u,  it  seems  somewhat  super 
fluous  to  expand  a\u  in  powers  of  u. 
From  the  formula  _ 

NV^=  °&, 

ou 
it  follows  that 


\—  -  ex 
[_u2 


ou— 

20 


or 


Methods  including  recursions  formulas  for  the  further  expansion  of  these 
functions  are  found  under  the  references  given  above.  In  particular 
attention  is  called  to  the  formulas  that  result  from  the  partial  differen 
tiations  with  regard  to  the  invariants  (given  by  Halphen,  loc.  tit.,  t.  I, 
Chapter  IX;  Frobenius  and  Stickelberger,  Crelle,  Bd.  92,  p.  311). 

EXAMPLES. 
1.    Show  that 

a(2  u)  =  2  °(u 


oa>  o<jL>a(jL) 

2.  Show  that 

o(u  +  v}  o(u  -v)        d2  .  d2     ' 

^  -  rf-j  -  '•    =  —  log  OU  ~  —  log  (7V. 

o2uo2v  du2  dv2 

3.  Prove  that  (if  a>^  aj^  a)v,  =  cu,  to"  ,  to'  without  respect  to  order) 

(1)  SIQ(U  +  wi) 

(2)  ?xo(u  +  cjj 

(3)  -u  +  an} 


(4) 


THE   SIGMA-FUXCTIONS. 
4.   Verify  the  formulas 

*  fu  +  v\=  ^oM-nM 


-  (ev  -  efi  (ev  -  ^)c20l/u)c20 


^(u  +  v) 
5.   Show  that 


m  =  x 

-fa  I1 


cos- 


UJ 


n 


1 


sin" 


6.   Show  that 


-nr- 


2  <$u  —  €)       O)U       ou      du        au 


(«.- 


2  (<u  — 


u       ou       du 


_^_ 

C/?i 


du2 


7.   Show  that 


.  fc) 


,  k)  .  E($,  k)       k    sin  0  cos 


395 


where  F(0,  /v)  and  ^(^  A*)  are  Legendre's  integrals  of  the  first  and  second  kinds; 
and  that 

fr)  _      F(^fr) 
dk 


, 
k  k 


fed  - 


'   -  (3  ft*  - 


dfc 


A-)  +  fe  A  fe  Sin  ^  COS  ^  •  0, 
,  A:) 


,  fc) 


,  A-)       A- 


A-  A-    sin  0  cos  0  _  n 

(^k]  ~ 


Write  <£  =  -  in  these  equations  and  note  the  results. 


CHAPTER   XVIII 

THE  THETA-  AND  SIGMA-FUNCTIONS  WHEN  SPECIAL  VALUES 
ARE  GIVEN  TO  THE  ARGUMENT 

ARTICLE  337.     The  theta-functions  were  expressed  in  Art.  209  through 
the  following  formulas  : 


m=l 


=  2      sn 


in  nu  JJ  (1  -  q2  m)  JJ  (1  -  2  g2  m  cos  2  TTM  +  ?4m), 


m=l 


2  gicosTTuJX  (1  ~  52m)  II  t1  +  2  52m  cos27rw  +  q4m), 

m=l  m=l 


m=l 


For  brevity  we  put 

m  =  oo 

Oo=  II  d  -  a2*). 


Since  these  quotients  are  absolutely  convergent  (Art.  17),  we  may  write 

QoQs  = 
and  consequently 


m=l 

It  follows  *  that 


*  See  the  16th  Chapter  of  Euler,  Introdutio  in  analysin  infinit. 

396 


TRANSCENDENTAL  CONSTANTS.  397 

Making  the  argument  equal  to  zero  in  the  theta-functions,  it  is  seen  that 


ART.  338.     From  the  following  formulas  (see  Art.  208), 

m  =  x  m  =  +  x 

1  +  2  2)  (-  I)mqm*cos2m-u  =  2)  (-  \)^e'm^ie 


-w  =  i    2)  (-l)me      * 


(2m+l)2     . 

~~ 


m  =  0 

m  =  x    (2m+l)2  m=  +  x    (2m+l)2 

2)  9      4      cos  (2  m  +  l)-w  =    2)  e      4 

m  =  0  m=  —x 


t?3(u)  =  1  +  2  2  2m*  cos  2  nant  = 

m=l 

we  have 

m  =  » 

#0=  1  +22)  (-  I)w9m=  1  -2 


m=l 


=  2-2)  (-  l)m(2m+  l)q      4       -  2-^  (1  -  3  q2  +  5  qQ-  7  q12  +  -   •   •) 


m  =  0 


(2m+l)2 

4 


=  x    (2m+l)* 

4 


m=0 

(2m+l)8 
>M  _I_  ^2^        4 


7M  =  30 

+2 


398  THEORY   OF  ELLIPTIC   FUNCTIONS. 

ART.  339.  Since  the  functions  $o>  &i,  $2,  $3  depend  only  upon  one 
variable  q,  it  is  natural  to  expect  that  they  are  connected  by  three  rela 
tions,  which  we  would  suppose  are  of  a  transcendental  nature.  Two  of 
these  relations  *,  however,  as  we  shall  show  in  the  sequel,  are  algebraic, 
viz., 


The  first  of  these  follows  at  once  from  the  equation  (Cf.  Art.  193) 

&2  +  fc/2=   1. 

To  derive  the  second  we  use  the  equation  of  Art.  295, 

#2#3#l(^  +  V)&0(U  -  V)  =  &i(u)&0( 

Expanded  in  powers  of  u,  it  becomes 


(  ) 

&M$*(v)  <#/#o  • u  +  ° '  ^2+  *  '  '  c 
I  ) 


0 


If  the  coefficients  of  u2  on  either  side  of  this  equation  are  equated,  we 
have  f 


an  expression  which  differentiated  with  regard  to  v  becomes 


If  we  put  v  =  0  in  this  equation,  we  have 


or 


^2  ^3 


*  They  are  both  due  to  Jacobi,  Werke  I,  pp.  515-17. 

t  See  Koenigsberger,  Ell.  Fund.,  p.  380;  or  Burkhardt,  Ell.  Funkt.,  p.  120. 


TKAXSCEXDEXTAL   CONSTANTS  399 


ART.  340.     It  may  next  be  proved  that 


a  r  4xi        du2 

Take,  for  example,  the  equation 


<x(u,  r)         (a=  0,1,2,3). 


When  differentiated  with  regard  to  r,  it  becomes 


_ 


dU2 

By  Maclaurin's  Theorem 


and  consequently  also 


dr  6r         2     dr 

If  these  values  are  substituted  in  (1),  we  have 


o  +  o      - 

dt        2     dr 


or  writing  u  =  0, 


In  a  similar  manner  it  may  be  shown  that 


/'=4;r  (,1  =  0,2,3), 

a- 


and  also  that 


Writing  these  values  in  the  last  equation  of  the  preceding  Article  and 
integrating  we  have 


If  both  sides  of  this  equation  are  expanded  in  powers  of  q,  it  is  seen  that 
the  constant  C  —  /r,  and  consequently  that 


It  is  also  seen  from  the  results  of  the  preceding  Article  that 


400  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

ART.  341.     If  the  formula 


_ 


be  differentiated  with  regard  to  u,  we  have 


en  u  dn  u  =  — ^ 

#2 


If  in  this  expression  we  put  u  =  0,  it  follows  that 


_J or     i  = 


It  is  thus  seen  that 
From  the  formula 
if  also  follows  that 


,      2KV2Kkk' 

[S     ~vT 


We  note  *  (see  also  Art.  345)  that 


or  since 
we  have 

and  also 


It  is  seen  that  k  and  kf  considered  as  functions  of  g  =  e1*1  are  one-valued 
functions  of  T.  From  this  point  of  view  Kronecker  found  the  origin  of 
some  of  his  most  beautiful  discoveries  and  Poincare  was  also  thus  led  to 
the  discovery  of  the  Fuchsian  Functions. 

*  See  Jacobi,  Werke  I,  p.  146. 


TRANSCENDENTAL   CONSTANTS.  401 


Hermite*  wrote  -\ik  =  <!>(T)  and  $&=+(*),  where  from  above 
and  T/r(r)  are  one-valued  functions  of  -  which  may  be  expressed  as  quotients 
of  two  infinite  products.  These  functions  are  of  such  importance  that  we 
may  consider  them  more  closely  and  at  the  same  time  introduce  other 
interesting  formulas  for  the  elliptic  functions. 

ART.  342.     From  the  equation 


m=l  m=l 

it  follows  that 


7n  =  l  m=l 

Since 

1  _  22(2m-l)  COS 


2  g2™-1  cos  2~u+ 

t?o(2  u,  9*>-n 

or 

(1) 
and  similarly 

(2) 

i 

We  also  have  from  the  product  of  &0(u,  q)  and  ^i(u,  5)  the  formula 


and  since 

1  -  2 
it  follows  that 


m  =  *  (I  -  o2m>)2 
q)&i(u,q}=q*Tl      l  _  \J 

771  =  1 

further  noting  that 

if  a  -  9m)=ii  a 

W=l  771  =  1 

we  have 

(3)  0o(M,g)#i(M,9)=  8*^^  v/9)- 

^3 

*  Hermite,  Resolution  del'  equation  du  cinquieme  degre.  (Euvres,  t.  II,  p.  7;  and 
also  Swr  /a  tfuorie  des  equations  modulaires,  CEavres,  t.  II,  p.  38;  see  also  Webber, 
EUiptische  Functionen,  pp.  147  and  327. 


402  THEORY   OF   ELLIPTIC    FUNCTIONS. 

If  for  u  we  write  u  +  J  in  this  equation,  it  becomes  (see  Art.  208) 
(4)  &B(u,  q)»2(u,  q)  =  9*       &2( 


If  for  q  we  write  qe**  =  -q  =  qe-*,  the  quantity  g*2o  becomes  q*e~*^ 

Qs  Q2 

and  the  equations  (3)  and  (4)  become 

' 


- 
(5)  &3(u,q)&i(u,q)=qle    8          i(ut  e2  Vq), 

Q2 


(6) 


The  six  formulas  above  are  given  by  Jacobi  (Seconde  memoire  sur  la 
rotation  d'un  corps.     Werke,  II,  p.  431). 
In  the  formula 


m=0 


the  summation  is  taken  over  positive  integers  including  zero.  If  we 
separate  the  even  integers  and  the  odd  integers  by  writing  m  =  2  n  and 
m  =  —  (2n  +  1),  we  have 


in  (4n 

n  =  —oo 

and  similarly 


Since 

sn 


Vk  ^o 
A; 


dn2Ku= 


it  follows  from  the  formulas  above  that 
(7) 


V2  V7^0(2  w,  q2)        ^k  ]     (  _  1)^2  n'  CoS  4 


TRANSCENDENTAL  CONSTANTS.          403 

where  the  summations  on  the  right  are  over  all  integers  from  n  =  —  oc  to 
n  =  -f  QC  .  The  summations  are  taken  over  the  same  integers  in  the 
following  formulas: 


(8) 


(10) 

(ID  c^wsv^^^ 

V  tf&^Vq)        • 


(12) 


in  (4  n 
If  we  put  u  =  0  in  (8)  and  (9),  we  have 


Jacobi  (Werke  II,  pp.  233-235)  has  given  several  different  forms  for 
these  two  quotients  of  infinite  series. 

If  we  write  u  =  0  in  (10)  and  (12)  and  determine  the  resulting  indeter 
minate  forms,  we  have  * 

2  \/2 


-  l)n(4n 


~  l)n(4n  +  I 

ART.  343.  By  equating  the  expressions  for  the  theta-f  unctions  in  the 
form  of  infinite  products  and  in  the  form  of  infinite  series  we  may  derive 
interesting  relations  connecting  the  quantity  q. 

For  example,  in  the  case  of  $i(w)  we  have  after  division  by  q* 

(1)     sin-w(l-52)(l-252cos2^  +  54)(l-54) 
=  sin-w  —  g2  sin  3  TTM  +  g6  sin  5  TTU  —  g12  sin 

*  See  Hermite,  (Euvres,  t.  II,  p.  275. 


404  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

If  in  this  equation  we  put  u  =  J  and  divide  by  i  V§,  we  have  * 


or  writing  q6  =  t,  it  follows  that 

m=oo  m=+Go  3m2+m 

(2)  jj(i  -  **)  =  2)  (-  i)»«  *'•  . 

w  =  l  m=  —  oo 

Upon  this  formula  depends  the  trisection  of  the  elliptic  functions. 

If  further  we  divide  equation  (1)  by  sin  nu  and  then  put  u  =  0,  we  have 

[(1  -  g2)(l  -g4)(l  -g6)   •    •   •  ]3  =  1  -3q2  +  5qQ-  7g12  +  9g20- 
Writing  g2=  Z  in  this  equation,  it  follows  that 


If  we  compare  the  equations  (2)  and  (3),  it  is  seen  (cf.  Jacobi,  Werke,  I, 
p.  237)  that 

(1  -  q-  £2+g5+g7_  ql2+  .    .    .)3=  !  _  3g  +  5g3 

Further  in  equation  (1)  put  Vq  in  the  place  of  q. 
We  then  have 


-  g2)(l  -  g3)   .  .  .  sin7rw(l  -  2qcos27tu  +  q2) 

(I  -  2q2  cos  2  7m  +  g4)  (1  -  2  #3  cos  2  KU  +  g6)  .  .  . 
=  sin  nu  —  q  sin  3  TTW  +  g3  sin  5  TTW  —  g6  sin  7  TT^  +   •  •  •   . 

Write  in  this  equation  u  =  J  and  observe  that 

QsQoQi2Q22  =       ;it  follows  that 


If  we  compare  the  two  expressions  for  &Q(U),  we  have 

Q0(l  -2q  cos  2xu  +  q2)  (1-2  g3cos  2  TTW  +  q6)  .  .  . 

=  1  —  2q  cos  2  7m  +  2  q4  cos  4  7m  —  2  g9  cos  6  ?m  + 
In  this  equation  write  u  =  0  and  observe  that 


It  follows  that 

---q*).    .   .       1      „  , 

~  29  - 


*  See  Euler,  Introductio  in  analysin  infinit.,  §  ^23. 


TRANSCENDENTAL   CONSTANTS. 
From  the  formulas 


405 


2Ku 


\  1  -  2q2m~1  cos2u  +  g4m' 

1  4-  2g2mcos  2u  +  g4m 
L^  1  —  2g2m-1  cos  2  u  +  q4m~ 


it  follows  that 
(1)     log  sn 


(2)     log  en  -     -  =  log  (2  q*  y  —  cos  i*j  +  2  2*  J~      4.  (_    \m  cos  2  mw' 


(3)     log  dn  ^^  =  log  Vk'  +  4 

From  (1)  and  (2)  we  have 
2  Kw' 


t2m-l 


^-;cos  (4m  —  2) 


sn 

log — r-1 
sin  fc 


i        ^  rv.        i         ^  (7*          p.    ^-\     i 

=  log =  log  -*=  +  2  >    — 


e» 


2  Ku 


log 


cos  z^  _ 


We  also  have  from  (1),  (2)  and  (3)  the  formulas 
(4)     Alogsn— =  ?- 


t^l^iKM 


m  =  x 


sn 


sin 


1  +  q 
i 


(5)    — T- 


cosu 


sin  2  mu, 


(6)    -^-logrfn^ 


2K 


2  KM      2  KM 

en 


O 

=  8 


, 
an 


;sin(4?tt  —  2)M. 


406 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


If  in  (4)  and  (5)  we  put  -  —  u  for  u,  we  have 
ft 


sn  u 


cn 


2Ku 


S1D 


2Ku 


71  71 

To  these  we  add  the  equations  of  Art.  231 


TT         2  Ku      sin 
sn  — 


=  -4-4 


--—  sin  (2m 
1  —  q2  m  ~  1 

=  l 


(10) 


en 


m=l 


and  the  equations  of  Art.  228 


--ain  (2m  _ 

=  1  ^ 


(13) 


cos2mu. 


In  Equa.  (12)  write  u  =  0  and  in  Equa.  (9)  put  u  =  -;  it  follows  that 


Y2m-l 


Similarly  writing  u  =  -in  (11)  and  u  =  0  in  (12),  we  have 


-i 


lm-1 


If  in  Equa.  (13)  we  put  -  —  u  for  u  we  have 


=  1+4 


m  =  l 


TRANSCENDENTAL   CONSTANTS.  407 

and  substituting  u  =  0  in  (10)  and  u  =  0  in  the  equation  just  written,  it 
is  seen  that 


+   n'2m 
,„  =  *  V. 

If  further  we  differentiate  (8)  with  regard  to  u  and  then  put  u  =  - ,  we 
have 


and  if  Equa.  (7)  be  differentiated  with  regard  to  u,  it  becomes  for  u  =  0 

m=« 

=  #4(0)  =1  +  8 


-        1 
Subtracting  (18)  from  (17)  we  have 


Jacobi  (Werke,  I,  pp.  159,  et  seq.)  has  given  forty-seven  such  formulas  as 
those  above. 

ART.  344.  In  Art.  89  mention  was  made  of  the  fact  that  many  of  the 
properties  of  the  0-functions  had  been  recognized  by  Poisson.  For 
example,  in  the  12th  volume  of  the  Journal  de  VEcole  Poly  technique,  p.  420 
(1823),  he  established  by  means  of  definite  integrals  the  formula 


tf- 


2  e~9}C/x  +  2  6~163r/ 


T^V 

To  verify  this  formula  by  means  of  the  elliptic  functions,  let  x  =  —  • 


-I  jr 

tity  x  becoming-  =  —  .     Hence  if  in  the  formula 

.''  L\. 


Instead  of  k  we  take  the  complementary  modulus  fc'  =  \/l  —  k2,  the  quan- 

'M 

v/^=1  + 

V    ^ 

we  change  k  to  &',  we  have 


_5          _  if 
=l  +2e   x  +  2e    * 


and  consequently  the  formula  of  Poisson.* 

*  In  this  connection  see  a  remark  by  Abel,  Crelle,  Bd.  4,  p.  93. 


408  THEORY    OF   ELLIPTIC   FUNCTIONS. 

ART.  345.     In  Arts.  260  and  320  we  derived  the  relations 
au  =  ?e2^2#iO),  [u 


where 


Noting  that 

we  have,  if  we  put 


.     *'»= 


e\—  63 


—  63 


If  follows  immediately  that 


/2o>  = 
*    n 


7T 


or 


/2co 
VT= 


—  63  — 


We  also  have 


TRANSCENDENTAL   CONSTANTS. 


2(1  4-  2 


409 


=  #2(0) 


2  > 

/ 


It  is  further  seen  that 


or 

and  similarl 


2(61  + 

92=  -(TT- 


t?28(0)  +  t?38(0)], 


»-5?f- 


=  4  e 


and 


=(e3- 


=  _. 

16    Wi2 

ART.  346.     The  formulas  of  the  preceding  Article  may  be  written 

(1) 


(2) 


(3) 


(4) 


T)  - 


or. 


(5) 


410  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Noting  that  the  coefficient  of  u3  in  ou  is  zero,  and  that  the  coefficient  of 
u2  in  oxu  is  —  J  e^  it  follows  by  a  comparison  of  the  coefficients  on  the 
right-hand  side  of  equations  (5)  and  (6)  that 


(8)  . 

2 

From  (7)  and  (8)  we  have  at  once  the  relation  of  Art.  339, 

'"  "  #2"(0)      fl3"(0) 

"  "  ' 


ART.  347.     The  formulas  of  Art.  329,  in  virtue  of  the  relations  just 
derived,  may  be  written 


The  six  formulas  of  Art.  328  thus  offer  a  means  of  deriving  the  values  of 
the  functions  o\,  <72,  <73,  having  as  arguments  the  quantities  oj,  o/',  CD'. 

The  results  as  set  forth  in  the  Table  of  Formulas,  XLIV,  should  be  veri 
fied.     We  have  for  example 


e  2 


Such  formulas  may  also  be  had  as  follows: 

Since  z  =  e***,  where  u  =  2  a>v,  when  -u  takes  the  values 

the  values  of  z  are  i,        iq*,        q*', 

and  since  W'—  in" to  =  — , 


7)0)  T)U          ,     Tl     n  , 

we  have  •&--  -    ^-  +  j  (1  +  r), 


so 


that  when  u  takes  the  values 


0) 


1)0)  Tj(l)  T)ll) 

2  2  2 


becomes          e,  e  iq±,        e 


TRAXSCEXDEXTAL  CONSTANTS.  411 

If  for  example  for  u  in  the  formula  for  on  (in  Art.  291)  we  write  u  =  u", 
we  have 


2i        l-g2 


' 


The  formulas  expressing  aiu,  a2u,  a$u  through  infinite  products  are  given 
in  Art.  321. 

EXAMPLES 
1.    Show  that 


(Jacobi,  Werke,  II,  p.  431.) 

2.   Through  a  comparison  of  the  coefficients  in  Formula  (6)  of  Art.  346  show 
that 


(a-  1,2,3;    #4 
3.   Show  that 

tei  —  e*)2      3 


~  27 


4.  Verify  all  the  formulas  given  in  the  Table  of  Formulas,  XLI  and  XLIL 

5.  Show  that 

4)3  27  <3  1    ^8(0)  +  #8(0)  + 


- 


CHAPTER  XIX 
ELLIPTIC  INTEGRALS  OF  THE  THIRD  KIND 

ARTICLE  348,     In  Chapter  VIII  we  saw  that  the  elliptic  integrals  of 
the  third  kind  in  the  normal  forms  of  Legendre  and  of  Weierstrass  were 

dz  C  dt 


r dz  and       f— 

J  (z2-  f?)V(l  -  z2)(l  -  /c2z2)  J  (t- 

In  the  neighborhood  of  the  point  z  =  /?,  if  ^  is  noZ  a  root  of 
S2  =  Z  (Z)  =  (l  -  z2)(l  -  /c2z2)=  0,  the  expansion  of 


v  (1  -  z2)(l  -  /c2z2) 
is  by  Taylor's  Theorem 


A 


-  z2)(l  -/b2z2) 
where 


1 


It  is  evident  that  Legendre 's  normal  integral  becomes  logarithmically 
infinite  for  z  =  /?  in  both  leaves  of  the  Riemann  surface  as  the  two  quan 
tities 

-LAlog(Z-0)     and     --LAlog(z-/?); 
&p  *  p 

and  for  Z  =  —  /?  in  both  leaves  as 

-J_Alog(Z+/3)     and     ^Alog(z  +  /9). 

If  /?  is  a  root  of  (1  -  z2)(l  -  k2z2)  =  0,  say  /?  =  1,  then  at  the  point  /?  =  1 
the  integral  becomes  algebraically  infinite  of  the  one-half  order. 

The  integral  of  the  third  kind  in  Weierstrass's  normal  form  becomes 
logarithmically  infinite  at  the  point  t  =  b  in  both  leaves  of  the  Riemann 
surface  as 


log(J-6)     and =logtf-6). 

v463-  gf2^>  -  93 
412 


ELLIPTIC   INTEGRALS    OF   THE   THIRD   KIND.  413 

ART.  349.  Let  us  next  form  the  simplest  integral  of  the  third  kind 
which  becomes  logarithmically  infinite  at  only  two  points  of  the  Riemann 
surface.  There  must  be  at  least  two  such  points  a\  and  a2>  say>  since  the 
sum  of  the  residues  of  the  integrand  must  be  zero. 

We  may  write  the  integrand  in  the  form 


(z-  «i)(z  -  «2)v2(z) 


We  shall  choose  the  points  [«i,  v/Z(a1)],  [a2,  x/Z(a2)  ]  in  the  upper  leaf  of 
the  Riemann  surface  and  we  must  determine  the  constants  A0,  A\,  A2 
so  that  the  integral  does  not  become  infinite  at  the  two  corresponding 
points  [a i,  —  \/Z(ai)],  [a2,  —  VZ(a2)]  in  the  lower  leaf. 
Accordingly  we  must  have 


(A0+  Aiai-  A2NZ(tti)  =  0, 

=  0. 


In  the  neighborhood  of  the  point  z  =  a\  we  have  by  Taylor's  Theorem 


(z-a2)VZ(z) 
and  consequently 


It  follows  that 

Res/(z,  s)=  ^° 


[which  owing  to  equations  (1)]  = 
In  a  similar  manner  we  have 


(ai—  o:2)  Z(di) 

2  1 


Res  /(z,  s)         O         ^2      A 


(«2-  ai)  v/Z(a2) 
Eliminating  J.x  from  the  equations  (1),  we  have 


and  eliminating  A0  from  the  same  equations,  we  have 


414  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  follows  that 


A2     a2\/Z(al]  -a1VZ(a2}  +  (VZfe)  -  v'Z(ai)  )z+  («2-«  i)V/Z(z) 
-  «i  (z  - 


=      A2     \     VZ(ai)WZ(z)  +  VZ(tt2)WZ(z)1 
«2-  ai  L       (z  -  «i)  VZ(z)          (z  -  «2)\/Z(z)  J 

When  7(z,  s)  has  this  form,  the  integral  /  /(z,  s)dz  is  of  the  third  kind,  being 

logarithmically  infinite  at  the  points  («i,  VZ(«i)),  (a2,  VZ(a2)). 

This  integral  may  be  considered  the  fundamental  integral  of  the  third 
kind  and  written 


Il(z,  VZ(z);  «i,  \/Z(«i);  «2>  VZ(^))  or  more  simply  II(z;  «i;  «2). 

In  a  similar  manner,  as  was  proved  in  the  case  of  the  integrals  of  the  second 
kind,  we  have  a  general  integral  of  the  third  kind  with  the  two  logarithmic 
infinities  «i7  «2  if  we  add  integrals  of  the  first  kind  to  H(z;  «i;  a2). 


ART.  350.     Take  three  points  ai,      Z(ai);  «27  VZ(«2);  «3j      Z(a3)  on 
the  Riemann  surface  of  Chapter  VI  and  form  the  integrals 

H(z;  ai;  a2),        n(z;  «2;  «3),       H(z;  «3;  o^). 

Further,  let  A2,  A2(1)  and  A3(1)  be  the  constants  that  correspond  to  A2 
above. 

We  may  so  choose  the  constants  A,  /*,  v  that  the  expression 

(1)  ^II(z;  «i;  a2)+/*n(z;  «2;  «3)  +  v  II(z;  o:3;  ai) 

does  not  become  infinite  at  any  point  of  the  Riemann  surface  and  is  con 
sequently  an  integral  of  the  first  kind  or  a  constant. 

We  note  that  in  the  neighborhood  of  the  point  «i  the  expression  becomes 
infinite  as 


a\—  ot-2  ai~  a 

and  consequently  remains  finite  at  a  i  if 


Similarly  the  expression  remains  finite  at  a2  and  a3  if 


+  =  0    and 

a2—  a\       a2—  «3  a3—  a2 


ELLIPTIC   INTEGRALS    OF   THE   THIRD   KIND.  415 

The  third  equation  is  a  consequence  of  the  first  two.  If  the  ratios  of  /,  p,  v 
have  been  determined  from  these  equations  the  integral  (1)  is  an  integral 
of  the  first  kind  *  or  a  constant. 

ART.  351.  We  have  seen  in  Chapters  VII  and  XIII  that  the  integral 
of  the  first  kind  has  in  common  with  the  integral  of  the  second  kind  the 
property  of  being  a  one-valued  function  of  position  on  the  Riemann  surface 
Tf.  This  is  not  true  of  the  integral  of  the  third  kind;  for  consider  in  the 
Riemann  surface  the  fundamental  integral  above.  In  the  neighborhood 
of  the  point  z  =  «i  we  saw  that  the  integrand  could  be  written  in  the  form 

P(z-  ai). 

It  follows  that  the  integral  over  a  small  circle  including  the  point  a\  as 
center  is 

2A. 


while  the  integral  over  a  small  circle  including  the  point  z  =  «2  is 

2i  7T1. 

If  then  two  paths  of  integration  (1)  and  (2)  starting  from 
the  point  po  include  both  or  neither  of  the  points  «i  and 
«2,  we  come  to  the  point  p  with  the  same  value  along  either 
path. 

Hence  to  construct  a  Riemann  surface  upon  which  the 
fundamental  integral  of  the  third  kind  will  be  one-valued 
we  draw  small  circles  around  a\  and  a2  and  join  these 
circles  by  a  canal  so  as  to  form  a  connected  curve.  To 
make  the  surface  simply  connected  we  join  this  canal  with  the  canal  «, 
say  in  Tf  (of  Art.  142),  by  another  canal  AB.  The  new  surface  we  denote 

by  T". 

Denote  the  difference  in  values  of  the 
integral  H(z;  a\\  a^}  taken  on  the  left  and 
right  banks  of  the  canals  in  T"  by  II(/)  — 
II((0).  It  is  seen  that  for  the  canal  AB  any 
path  of  integration  must  encircle  both  or 


neither  of  the  points  «i  and  «2  to  get  from 
the  left  to  the  right  bank.     It  follows  that 
Fig>  74>  along  the  canal  AB  we  have  H(X)  -  U(p)  =  0. 

*  See  Clebsch  und  Gordan,  Theorie  der  Abel'schen  Functionen,   p.  118;  or  Klein- 
Fricke,  Theorie  der  elliptischen  Modulfunctionen,  Bd.  I,  p.  518. 


416  THEORY   OF   ELLIPTIC   FUNCTIONS. 

^  To  go  from  the  point  D  to  the  point  C  in  the  figure  we  must  encircle 
either  a\  or  a2.     In  'either  case  we  have 


This  difference  may  be  made  -  2  m  if  in  the  fundamental  integral  we  give 
to  the  arbitrary  constant  A2  a  value  such  that 


A*  1 


0.2  —-   Oil         2 

ART.  352.     Let  us  consider  next  the  elementary  integral  of   the   third 
kind  in  the  Weierstrassian  notation 


;    a,  VS&);    00)  =    C" 
J 


+ 


2  (t  -  a)         VS(t) 
where  S(t)=  4  Z3  -  g2t  -  gs. 

Writing  /?  =  VS(a)  we  note  that  in  the  neighborhood  of  the  point  (a,  /?) 
we  have 


so  that  in  the  neighborhood  of  t  =  a 

U(t-  a;  oo)  =log(J-  a)-  ±  ^  (t  -  a)  +    •   .  - 

2ft  da 

and  that  the  residue  corresponding  to  the  point  t  =  a  is  +2  m. 
In  the  neighborhood  of  the  point  at  infinity  we  have 


1        _  1       a       a2 

~~  T  "i~  To  "i    ~TT~ 


-  «         t 


S(t) 
and  consequently  in  the  neighborhood  of  infinity 


ELLIPTIC    INTEGRALS    OF   THE   THIRD   KIND.  417 

Further,  if  we  put  t  =  reie 

and  v  =y  -  =  pe**, 

we  have  p  =  V  /  ->        <£  =  —  i  6, 

so  that  a  double  circle  about  the  point  at  infinity  in  the  ^-plane  corresponds 
in  the  opposite  direction  to  a  single  circle  taken  around  the  origin  in  the 
r-plane.  Hence  (see  Art.  120)  the  residue  corresponding  to  the  point  at 
infinity  is  —  2  TTI. 

ART.  353.  It  is  also  seen  that  if  in  the  T'-surface  we  draw  canals  from 
the  points  a\  and  infinity  to  the  canal  «,  say,  we  form  another  simply 
connected  surface  T"  in  which  the  integral  II(£;  a;  x)  is  one-valued.  On 
the  first  of  these  canals  we  have 


and  on  the  second  H(A)  -  TL(p)  =  —  2  xi. 

If  the  point  a  coincides  with  one  of  the  branch-points  ei,  say,  then  in 
the  neighborhood  of  t  =  e\  the  integral  11(2;  e\\  oc)  becomes  infinite  as 
log  \^t  —  e\]  while  in  the  neighborhood  of  t  =  oc  this  integral  becomes 
infinite  as  log  \/t. 

Further,  if  we  put 

II(*;  «2;  ai)=  H(t;  «2;  oc)-  Ufa  ttl;    oo) 


CVS(t)  +  VS(a2)       dt  C 


\/S(t)+VS(al)       dt 


it  follows  from  Art.  349  that  II(£;  «2;  «i)  becomes  logarithmically  infinite 
at  the  arbitrary  points  a2>  «i  but  has  a  definite  value  f  or  t  =  oc  .  If 
here  the  point  a\  is  in  the  lower  leaf  directly  under  a2,  so  that  a2  =  a\, 
=  —  Vo(«i),  then  the  above  integral 


<*i)\'S(t) 
which  is  the  integral  cited  at  the  beginning  of  this  Chapter. 

ART.  354.  To  study  the  moduli  of  periodicity  of  the  integrals  of  the 
first,  second  and  third  kinds,  Riemann  *  took  two  functions  u  and  v  and 
considered  the  integral 

J  UdzdZ' 

When  u  and  v  are  integrals  of  the  first  and  second  kinds  the  integrand  u  — 

dz 

is  one-valued  in  the  Riemann  surface  Tr ;  when  one  of  these  functions  is  an 

*  Riemann,  Theorie  der  Abel'schen  Functionen;  see  also  Appell  et  Goursat,  Fonctions 
algebriques,  Chap.  III. 


418 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


integral  of  the  third  kind,  the  integrand  is  one-valued  in  the  surface  T" . 
Riemann's  mode  of  procedure  is  essentially  the  following  :  The  integra 
tion  is  taken  first  over  the  entire  boundary  of  the  simply  connected  sur 
face  in  which  the  integrand  is  one- valued,  and  secondly  over  a  curve  which 
gives  the  same  value  of  the  integral  as  the  first  curve;  for  example,  the 
circle  or  double  circle  around  the  point  at  infinity.  Since  the  latter  curve 
contains  in  general  no  discontinuities  of  the  integrand,  the  associated 
integral  is  zero. 
Consider  the  two  integrals 


where  u  and  £u  are  integrals  of  the  first  and  second  kinds  respectively  and 
where  the  integration  is  over  the  complete  boundary  of  the  surface  Tf 
taken  in  the  positive  direction. 

Let  the  moduli  of  periodicity  of  II(£;  a;  oo)  on  the  canal  a  be  II (X)  — 
!!(/>)=  2  Uj  and  on  the  canal  b  let  H(p)  -  II (A)  =  2  u'.  Further,  note  that 
the  integrands  of  both  I\  and  1 2  are  continuous  at  the  point  t  =  GO. 

ART.  355.     The    Riemann   surface 
T"  projected  on  the  w-plane  is  (see 
Art.  197)  Represented  in  the  figure. 
It  is  evident  that 


\ 


Fig.  75. 


or 


U°udU+  fudtt  , 

uz  *J  around  a. 


CU\udTI-(u 

9S  UQ 


UQ 

=  -  2  oj' 


2iriu(a) 


2co 


+  2  niu(a) 


on  6 


2  niu(a). 


This  integral  around  the  circle  at  infinity  is  zero.     It  follows  that 


and  similarly  from  I2, 


ELLIPTIC    INTEGRALS    OF   THE    THIRD   KIND.  419 

Noting  that  TJO/—  cur)'  =  ^, 

it  is  seen  *  that  o  =  yu(a)  —  co^(a), 


The  quantities  u  and  u't  the  moduli  of  periodicity  of  the  integral  II  (Z;  a;oo), 
have  values  that  are  the  negative  of  those  given,  if  the  canals  a  and  b 
are  crossed  in  the  opposite  direction,  or  what  is  the  same  thing,  if  the 
direction  of  integration  around  these  two  canals  is  taken  in  the  opposite 
direction. 

EXAMPLES 

1.  Derive    the    results    analogous    to    those    given    above    for    the    integral 
II(z;  «1;a2),  the  surface   T"  being  that  given  in  Art.  351   (see  Forsyth,  Theory 
of  Functions,  §238). 

2.  Let  u  =  II(z;  at;  «2)  and  v  =  H(z;  «3;  «4)  and  discuss  the  moduli  of  periodic 
ity  in  the  associated  Riemann  surface  (see  Koenigsberger,  Ellip.  Funct.,  p.  278). 

ART.  356.     We  wrote  (see  Art,  196)  t  =  pu,  VS(f)  =  -  p'u;  it  follows, 
if  a  =  $>UQ,  VS(a)  =  —  P'UQ,  that 


•V8®VS(t)+  VSM       dt      _  I   ruy'u  4-  9'\i 
2(t  -  a)          V~S(?)       2J      V"  ~  fi?M- 


The  quantity  UQ  must  not  be  congruent  to  the  origin.     In  Art.  299  we 
saw  that 

1  p'u  +  p'lip  =  a'(u  -  Up)  _  o^u        O'UQ_ 

2  $>U  —  $>UQ  o(ll   —   UQ)  OU 

Through  integration  it  follows  that 

0(u0  -  »)  ^ 


1    (-"<?'  u  +  V'tlo  du  = 

2J  $11     —     tfllQ  OUOUQ 


The  constant  C  is  to  be  so  determined  that  in  the  development  (see  Art. 
300) 

l  r-g^±^dtt  =  _  iogtt  +  c- 

2  J      pu  -  <?uo 
C  is  zero. 

We  then  have  (see  Schwarz,  loc.  cit.,  §  56) 

U(t;  a;  oe  )  =  log  a(u°~  u)  +  u 


*  See  Schwarz,  loc.  cit.,  §  59. 


420  THEORY   OF   ELLIPTIC    FUNCTIONS. 

It  follows  at  once,  if  m  is  an  integer,  that 


',a;  oo)-  II(a;$;oo)  =  u         -  -  UQ  —  +  (2m  +  l)m, 

(JUQ  OU 

a  result  which  corresponds  to  the  interchange  of  argument  and  parameter 
in  the  Legendre-Jacobi  theory  of  Art.  258. 

ART.  357.     Legendre  Traitc  des  fonctions  elliptiques,  t.  I,  p.  18,  repre 
sented  the  elliptic  integral  of  the  third  kind  in  the  form 

II  (n,  fc,  <«  =   C-  -  ^  [see  Art.  167], 

J  (1  •+  nsm2(>)  A<> 


where  the  parameter  n  may  be  positive  or  negative,  real  or  imaginary. 
This  integral  may  be  written 

TT/     7      \       C        du 
U(n,k,u)  =   I  -—    —-- 
J  1  +  n  snzu 

It  follows  that 


TT/     7      \  C  —n  sn2u    -, 

tt(n,k,u)-u  =   I—- —  du, 

J  1  +  n  snzu 


where  u  is  an  elliptic  integral  of  the  first  kind.     Jacobi  [Werke,  I,  p.  197] 
made  a  further  change  in  notation  by  writing  [see  also  Legendre,  loc.  cit., 

p.  70] 

n  =  —  k2sn2a, 

where  a  being  susceptible  of  both  real  and  imaginary  values,  leaves  n 
arbitrary. 

J      Q  -t 

Multiplying  the  right-hand   side  by  —  >  the  form  of  the  elliptic 

STL  a 

integral  of  the  third  kind  adopted  by  Jacobi  is 

uk2sna  cna  dna  sn'2u  -, 


ART.  358.     In  Art.  294  the  following  equation  was  derived: 

92(0)  ecu  +  a)  ecu  -a)  =  l  _  k2sn2u  Sn2at 


If  we  differentiate  logarithmically  with  regard  to  a,  we  have 

2  k2sna  cna  dna  sn2u  _  ®'(u  —  a)  _  Q'(u  +  a)    ,    pQ^a) 
1  -  k2sn2u  sn2a       ~  ®(u  -  a)        ®(u  +  a)  0(a) 

from  which  it  follows  at  once  that 

TT,       v      1,      ®(u  -  a)    .      0'(a) 


ELLIPTIC    INTEGRALS    OF   THE    THIRD    KIND.  421 

Interchanging  u  and  a  we  further  have 

n(a,u)-llo8e<*-M>+ag^, 
2     50(a  +  -w)  0(u) 

from  which  it  is  seen  that 

U(u,a)-  H(a,  u}=  uE(a)-  aE(u). 

We  note  that  this  equation  remains  unchanged  when  the  argument  u 
and  the  parameter  a  are  interchanged  (see  Legendre,  loc.  cit.,  pp.  132 
et  seq.). 

ART.  359.  ',  It  is  evident  from  the  integral  above  through  which  II  (u,  a) 
is  denned,  that 
(1)  H(M,  a)  =  -  H(-  u,  a)     and 

(2)  n(o,o)=o. 

Further,  since  snK  =  I,  en  K  =  0,  dnK  =  k',  we  have 

(3)  IL(u,K)=Q. 

For  a  =  iK'  we  have  sn  a  =  ao  =  en  a  =  dn  a,  so  that 

(4)  no/,  ;£')  =  «; 

and  since 

sn(K  ±iK'}  =  -,     cn(K±iK')=  ^F~t     dn(K  ±  iK')=  0, 
A;  k 

it  follows  that 

(5)  U(u,K  ±iK'}=  0. 

From  the  formula  expressing  the  interchange  of  argument  and  parameter 
we  have 

(6)  U(K,a)=  KE(a)-  aE=  KZ (a)     [Legendre]. 

These  formulas  follow  also  directly  from  the  expression  of  II (u,  a)  through 
the  theta-functions,  as  do  also  the  formulas 

(7)  IKK  +  iK',  a)  =  (K  +  iK')  Z(a)  + 


ma 


2K 

(8)  11(2  iK1,  a)  =  2  iK'Z(a)  +  |^, 

(9)  U(u  +  2K,  a)  =  II(M,  o)  +  2  KZ(a), 

(10)  U(u,  a  +  2K)=  n(u,a)  =  U(u,a  +  2iK'), 

(11)  n(M  +  2iK',o)=  H(M?O)+  2U(K  +  ^',0)-  2H(K,a) 

-  U(u,a)+2iK'Z(a)+  ^- 

A 

From  the  equations  (9)  and  (11)  it  is  seen  that  the  moduli  of  periodicity 
of  II  (u,  a)  are  respectively 

2  K Z  (a)     and     2  iK'Z  (a)  +  ^- 


422  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

ART.  360.     From  the  definition  of  U(u,  a)  given  in  Art.  357  we  have 

d  II  (u,  a)  _  k2sna  cnadna  sn2u 
du  1  -  k2sn2a  sn2u 

=  Z(o)  +  i  Z(u  -  a)  -  J  Z(u  +  a)     [Art.  297]. 

We  therefore  have  the  theorem:  The  derivative  of  an  elliptic  integral  of  the 
third  kind  with  regard  to  an  elliptic  integral  of  the  first  kind  may  be  expressed 
through  elliptic  integrals  of  the  second  kind. 

Interchanging  u  and  a,  we  also  have 

V" 
k2snucnudnusn2a       „  .          , 


The  addition  of  these  two  equations  gives 

Z(u)  +  Z(o)  —  Z(u  +  a)  =  k2snu  sna  sn(u  +  a), 

which  is  the  addition-theorem  of  the  Z-f  unction  (see  Art.  297). 
ART.  361.     From  the  formula 


Vk' 
we  have  by  writing  in  in  the  place  of  u 


V        ^«    v-  —  —   /  / 

Vk' 
or,  (see  Arts.  204  and  220) 


0(0,  k') 

If  we  take  the  logarithmic  derivative  of  this  equation,  we  have 
tZ(tu  +  K)  =  ^-r  +  Z(«  +  K',  k'). 

If  these  expressions  are  written  in  the  formula    • 

U(iu,  ia  +  K)=  iuZ(ia 


0^a  +  ^u  +  K) 
we  have 


H(iu,  ia  +  K}  =  TL(u,  a  +  K',  kf). 


ELLIPTIC    INTEGRALS  .  OF  THE   THIRD   KIND.  423 

If  a  is  changed  into  ia,  it  follows  that 

H(iu,  a  +  K)=-U(u,ia  +  K',  k'). 

These  results  may  be  derived  directly  by  a  consideration  of  the  integral 
which  defines  H(w,  a)  [see  Jacobi,  Werke  I,  p.  220]. 
ART.  362.     In  Art.  227  we  saw  that 


It  follows  directl    from  the  formula 


that 

Ku    2Ka\     2Ku 


n          ~ 


,   q  cos  2  (it  +  a)  _^  q2  cos4(u  +  a)  _, 
1-92  2(1  -g«) 

_  gcos  2(u  —  a)  _  q2  cos4(i£  —  a)  _ 
1  -  q2  2(1  -  54) 


2Ku 


Ku       \   -    I  _  9  |"g  sin  2  a  sin  2  u  q2  sin  4  a  sin  4  u 
~            2Ka           I        !-52  2(1  -5^) 

q3  sin  6  a  sin  6  u  .  "] 

3d  -96)  "J 


THE  OMEGA-FUNCTION. 
ART.  363.     Jacobi  (Werke,  I,  p.  300)  put 


/    E(u)du  =  log  li(w). 

i/O 


If  we  integrate  the  formula  of  Art.  297 

E(u  +  a)  +  E(u  -  a)  =  2  E(u)  - 

1  —  k2sn2a  sn2u 

from  u  =  0  to  u  =  u,  we  have  at  once 


log  -       +  iog         -       =  2  log  o(M)  +  log  (1  -  k2sn2a  sn*u), 

O(aj  O(a) 


or  O(M  +  a)  fl(?<  —  a)       1       790        o 

—  -  -  -  —  {  —  J  —  -  —  -  =  1  —  k2sn2a  sn2u. 


424  THEORY   OF    ELLIPTIC   FUNCTIONS. 

Further,  if  u  and  a  are  interchanged  in  the  above  formula,  it  becomes 

E(u  +  a)  -  E(u  -  a)  =  2  E(a)  -  2  k2sna  cna  dna  sn2u  ^ 

1  —  k2sn2a  sn2u 
which  integrated  from  u  =  0  to  u  =  u  is 


In  Art.  251  the  following  formula  was  derived: 

E(iu)  =  i  [tn(u,  k')  dn(u,  k')  +  u  -  E(u,  k')]. 
We  have  at  once 

n 

log  0(m)  =  log  cn(u,  k')-~  +  log  Q(u,  k'), 

ft 

or  _«2 


e    2  cn(u,  k')  Q(w,  A;'). 
ART.  364.     From  the  formula  E(u  +  2  mK)  =  E(u)+  2  mE  we  have 

f|^ 

n(u  +  2mK)=        2mE 
0(2  mK} 

If  we  put  M  =  —  2  mK  in  this  formula,  and  note  that 

0(-  u)  =  Q(u),     0(0)=  1, 
we  have  O(2  mK)  =  e2m'EK, 

and  also  to(u  +  2 

Or  - 


0(w  +  2mK)=  e    2V(u). 

Eu* 

This   formula   shows   that   the   function  e    2K  &(u)    remains   unchanged 
when  the  argument  is  increased  by  the  real  period  2  K. 
Further,  if  in  the  formula 


e    2  cn(u,  k')tt(u,  k'), 
we  write  u  +  2  nK'  in  the  place  of  u,  we  have 


(u+2nK'}2 


Q(iu  +  2niK')  =  (-  l)ne  cn(u,k')Q(u  +  2nK',k'), 

or 


e    2  n(iu  +  2niK')  =  (-l)ne          2       cn(u,k')e    *K'  Q(utW). 


ELLIPTIC    INTEGRALS   OF    THE   THIRD   KIND.  425 


It  follows  that 


~r(          ^ 


=  (-  l)*e    2K'       O(tu). 
If  in  this  expression  we  put  —iu  for  u  or  u  for  iu,  we  have 


6  Qw 

from  which  formula  it  is  seen  that  the  expression 


e      2K> 


remains  unchanged  when  u  is  changed  *  into  u  +  4 
ART.  365.     We  derived  in  Art,  263  the  formula 


ei—  e3      s 
from  which  we  have  at  once  through  logarithmic  integration 


-  e3  •  u) 
Writing  these  values  in  the  formula 


, 

2        O(«  +  a) 

it  is  seen  that 


•  u,  ^.  a)  =  I  log          e^^u 

2       n[vei-  e3(u  +  a)J 

-  e3  •  a) 


2        <T3(w  +  a)  a3a 

[See  Schwarz,  loc.  tit.,  p.  52.] 

ART.  366.     The  following  relations  may  be  derived  from  the  addition 
theorems  of  the  theta-functions  given  in  Art.  211,  formulas  [C]: 

e«(0)H(n  +  a)H(u  -  a)  _      2    _      2 


»  +  a)H.  (u  -  a)  = 


*  See  Jacobi,  Werke,,  I,  p.  309. 


426  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

If  as  in  Art.  358  these  expressions  are  differentiated  logarithmically  with 
regard  to  a  and  integrated  with  regard  to  u,  the  variable  in  the  first  equa 
tion  being  less  than  the  parameter  a,  we  have 


r» 

Jo 


I 
r 


8nacnadnad  1  ,      H(q  -  u) 

o    sn2u  -  sn2a  2     to  H(a  +  u)  9  (a) 

uk2snacnadnacn2u  d  =l^      ®i(u  -  a)  0'(a) 

k2cn2ucn2a  +  k'2  2     g  ®i(u  +  a)  '  U  0(a) 


acnadna  dn2u  ,     =  1  j      Hi(t6  —  a)    ,      0'(cQ 
dn2udn2a  —  k'2  2        HI(W  +  a)  0(a) 


These  integrals  *  may  all  be  expressed  through  the  integral  II  (u,  a)  and 
an  elliptic  integral  of  the  first  kind ;  for  example 


f 

Jo 


sn  a  en  a  dn  a  i    _  TT/          ,    '/TM—  ucnadna 


sn  a 


ADDITION-THEOREMS  FOR  THE  INTEGRALS  OF  THE  THIRD  KIND. 

ART.  367.     The  addition-theorem  for  the  elliptic  integral  of  the  third 
kind  follows  directly  from  the  equation  of  Art.  358  in  the  form 


TT/       \      TT/       \     TT/  \       IT      ®(u  —  a)  0  (v  —  a 

ft(u,a)+  n(tva)-H(«  +  v,a)  =  -log  -)  --  (r>/    .       rv     . 

2        @(w  +  a)@(v  +  a)@(w  +  'y  —  a) 


For  brevity  we  shall  put 

+  v  +  a)  =  p(  } 

+  v  -  d) 

and  we  shall  derive  several  different  forms  for  F(u,  v,  a)  which  are  due  to 
Legendre  and  Jacobi.f 
From  the  formula 


02(0)  0(^  +  v)  0(jM  -  v)  =  02(/£)  ©2( 
we  have  at  once 


*  See  nofe  by  Hermite  in  Serret's  Calcul,  t.  II,  p.  840. 

t  Legendre,  Fond.  Ellipt.,  t.  I,  Chap.  XV;  Jacobi,  Werke,  I,  pp.  207  et  seq. 


ELLIPTIC    INTEGRALS   OF   THE   THIRD   KIND.  427 

and  by  taking  the  product  of  the  first  and  fourth  of  these  equations 
divided  by  that  of  the  second  and  third  we  have 


o/U  +  V          \1L        ,  0       2^  +  ^        2/U  +  V   i_     \1 

t2  — a      1-Fsn2-— sn2  — -  +  a) 

*•(",  *,  a)  =] x^  7       V    2 ({J ? ^ /I 

1  - A;2sn2 /'^^-Jsn2  f ^^  +  a J 111  - k2m2 ^^sn2(^--a 


From  the  formula 

,  x      /  x         sn2u  —  sn2v 

sn(/jL  +  v)  sn(/i  -  v)  = f-— — 

1  —  k^sn^juisn^v 

we  further  have 


—  V 


Taking  the  products  of  these  two  equations  each  multiplied  by  —k2  and 
adding  a  common  term  on  either  side,  we  have  * 


2          \    2 

multiplied  by  { 1  —  k2sn asnusnvsn(u  +  v  —  a] 


2  \    2 

•^-^•Sf^f^-a) 


Writing   —a   for  a  in   this  equation,  we  have  a  second  equation,  which 
divided  by  the  first  gives 


-  a 


1  +  k2sn  a  sn  u  sn  r  sn(u  -t-  r  -L  q) 
1  —  k2sn  a  sn  u  sn  v  sn(u  +  r  —  a) 

*  See  Cayley,  Elliptic  Functions,  p.  159. 


428  THEORY   OF   ELLIPTIC    FUNCTIONS. 

If  a  is  changed  to  —a  in  this  expression,  it  is  seen  that 

p,          ^_  1  —  k2sn  a  snusnv  sn(u  +  v  —  a) 
1  +  k2sn  a  snusnv  sn(u  +  v  +  a) 

ART.  368.     It  follows  also  from  the  expressions  given  in  the  preceding 
Article  that 

_  a)02(,  _  a)_  02(0) 


1  —  k2sn2(u  —  a)  sn2(v  —  a) 
u  +  a)0*(*  +  a)  =  0*(0)  . 


1  —  k2sn2a  sn2  (u  +  v  —  a) 

M  +  „  +  „)  =  e*(0)   e(«  +  »)e(«  +  P  +  2«)  . 
1  —  k2sn2a  sn2(u  +  v  +  a) 
From  these  equations  we  have 

F(u  v  a)=  H  *  ~  k2sn2(u  +  a)  sn2(?;  +  a)  }  {  1  -  k2sn2a  sn2(u  +  v  -  a)  }"P 
U  1  -  k2sn2(u  -  a)  sn2(v  -  a)  }  {  1  -  k2sn2a  sn2(u  +  v  +  a)  }  J 

ART.  369.     Since  II  (u,  a)  —  II  (a,  u)  =  wZ(a)  —  aZ(u),  we  have 
H(w,  a)  +  IL(u,  b)  -  n(^7  a  +  6) 

=  H(a,  w)  +  H(6,  w)  -H(a  +  6,  w)  +  w{Z(a)  +  Z(6)  -  Z(a  +6)  } 
=  J  log  F(a,  6,  u)  +  w  fc2sn  a  sn  b  sn(a  +  b), 

which  is  a  theorem  for  the  addition  of  the  parameters. 
ART.  370.     In  the  formula  (see  Table  (B)  of  Art.  211) 

y)  =  &(x  +  y 
y  +  «)^i(«)^i 


write       3.  .  2^?      =  2&  and  ^  =  _  2Ka  and  +  2Ka  respectively. 

71  71  7T  7T 

Divide  the  first  result  by  the  second  and  we  have 

l  _  H(o)H(M)H(p)H(M  +  v-a) 

®(u  -  a]  ®(v  -  a)  ®(u  +  v  +  a)  _         @(o)  0(u)  Q(t;)  Q(i^  4-  y  -  a)  ; 
0(w  +  a)  0(v  +  a)  @(w  +  v  -  a)  ~  -        H(o)H(M  ' 


F(          x  _  1  —  fe2sna  snu  snv  sn  (u  +  v  —  a) 
1  +  k2sna  snu  snv  sn  (u  +  v  +  a) 

Remark.  —  By  writing  as  we  have  done 

n  =-k2sin26, 

and  allowing  6  to  take  imaginary  values,  the  expression  on  the  right-hand 
side  of  the  addition-theorems  above  is  always  a  logarithm.     Legendre  * 

*  Legendre,  Traite  desf  auctions  elliptiques,  t.  Ill,  p.  138. 


ELLIPTIC   INTEGRALS    OF   THE   THIRD   KIND. 


429 


considered  the  following  two  cases,  to  the  one  or  the  other  of  which  by 
means  of  real  transformations  the  parameter  n  may  always  be  reduced: 

(1)   n  =  -  k2sin20,     (2)  n  =  1  +  k'2sm26, 

where  6  is  real  in  both  cases. 
Owing  to  the  fact  that 


tan-1  it  =  -i 
2 


1  —  t 


the  inverse  tangent  appears  in  the  second  case  instead  of  the  logarithm.* 
ART.  371.     If  we  put 


we  have  from  Art.  355 

U(tl;  t0; 


OU\OllQ 


t0;  x)= 

t0;  *)= 
If  u%  =  ^1+  u2,  it  follows  that 


;   oc)-  log/  (i^3,  112,1*!,  u0), 


where 


Olif\— 


2 


P'UO) 
Wo  ) 


The  last  formula  is  verified  by  using  the  equation  (see  Art.  335,  [B.]) 
owa(u  +  v  +  w)  0i(u  —  v)  =  o(u  +  w)  a(v  +  w)  o#i  aw  —  oi(u  +  w)  a),(v  +  w)  auaVj 


and  the  formulas  given  in  the  Table  of  Formulas,  No.  LXII,  combined  with 
the  formula 


7, 

-^- 

02U 


It  follows  that 

;  t0',   oo) 


^2-^0 
[See  Schwarz,  loc.  tit.,  p.  90.] 

*  As  an  application  of  Abel's  Theorem,  Professor  Forsyth  (Phil  Trans.,  1883, 
p.  344)  has  given  a  very  elegant  method  for  the  addition  of  the  elliptic  integrals  of 
the  third  kind.  See  also  a  paper  by  Rowe  (Phil.  Trans.,  1881,  p.  713). 


430  THEORY   OF   ELLIPTIC    FUNCTIONS. 

EXAMPLES 
1.   Show  that 

IL(u  +  %K,%K}  =  i(l  -  k')(u  +  \  K)  -  i  log  dn  u  +  $  logX/Jfc7, 


U(u  +  i  itf',  i  itf')  =  i  i(l  +  k)  (u  +  \  iK'}  -  i  log  «n  M  +  i  log    - 

V/c 


iX')  -  4  log  en  w+  i  log 

A; 


.   Show  that 

H(M  +  K,  a)  =  H(M,  a)  +  KZ(d)  +  i  log 


a) 


II(it,  a+  A)  =  II(w,  a)  —  k2  sna  sin  coam  a  •  u+  %  log  —  -  -  * 

dn(u+  a) 


3.   Verify  the  formulas 

du_  _      dlo 

~  da  2~*®(u-a) 

£^)  +  IlogH(! 
da  2        H(u  —  d) 

_M<Oog©M__Llog< 

—  k2sn2(ia)sn2u]  da  2i        ®(u  —  id) 


rdna  cot  am  a  du  _     d  log  H(a)  _  1 ,      ®(u  +  a) 
1  -  k2sn*a  sn2u  ~  ' 

CK  sna  cna  dna  du  =  , ^  _    .  d  log  0 (a)       1 ,      H(it  +  a) 
Ju        sn2u  —  sn2a 


rfc2sn  (10)  en  (id)  dn  (id)  sn2u  du  _         d  log  @  (la) 1_ ,      G(M  +  i'a) 


4.   Show  that 

H(u,  d)  +  H(v,  a)  -  TL(u  +  v,  a) 

2         £l(u  +  a)  O(v  +  a)  O(w  -f  v  —  a) 
and  that 


-  a)  O(M  +  v+  a) 


(w  —  a)  O(v  —  a)  O(w  J2  /  \ 

i-^ L  =  1  —  k*  sna  snu  snv  sn(u+  v  —  a), 


CHAPTER   XX 

METHODS  OF  REPRESENTING  ANALYTICALLY  DOUBLY  PERIODIC 
FUNCTIONS  OF  ANY  ORDER  WHICH  HAVE  EVERYWHERE 
IN  THE  FINITE  PORTION  OF  THE  PLANE  THE  CHARACTER 
OF  INTEGRAL  OR  (FRACTIONAL)  RATIONAL  FUNCTIONS 

ARTICLE  372.  We  have  seen  that  the  simplest  doubly  periodic  functions, 
which  in  the  finite  portion  of  the  plane  have  everywhere  the  character  of 
integral  or  (fractional)  rational  functions,  are  the  functions  pu,  snu,  etc. 
We  shall  show  in  the  present  Chapter  that  all  other  doubly  periodic  func 
tions  which  have  the  properties  just  mentioned  may  be  expressed  in  terms 
of  these  simpler  functions. 

We  shall  study  in  particular  five  kinds  of  representations: 

(1)  Representation  as  a  su?n  of  terms  each  of  which  is  a  complete  derivative. 

(2)  Representation  as  a  rational  function  of,  say,  $>u  and  p'u  [Liouville's 
Theorem]. 

(3)  Representation  in  the  form  of  a  quotient  of  two  products  of  theta- 
functions  or  sig ma-functions. 

(4)  Representation  in  the  form  of  a  sum  of  rational  functions. 

(5)  Representation  in  the  form  of  a  sum  of  rational  functions  of  an  expo 
nential  function. 

ART.  373.  The  first  representation  mentioned  above  and  due  to  Her- 
mite  has  been  made  fundamental  throughout  this  treatise;  upon  it,  as 
already  stated,  the  other  representations  all  depend.  We  shall  produce 
it  again  in  a  somewhat  different  form  so  that  the  dependence  upon  it  of 
the  other  representations  may  be  more  readily  seen.  In  Art.  87  Hermite's 
intermediary  function  of  the  first  order  was  denoted  by  X(w)  and  was 
defined  through  the  equation 

m  =  + »  2ri'mu  .  6 

X(M)  = 


We  saw  that  this  function  satisfied  the  functional  equations 
(l)    X(t*  +  a)-X(iO, 


--(2u+6) 
(2)       X(M  +  &)=?      ° 


We  also  saw  that  this  function  vanished  on  the  point  -~ —  =  c  and  on  all 
congruent  points,  but  nowhere  else. 

431 


432  THEOEY   OF   ELLIPTIC   FUNCTIONS. 

By  writing  X(w  +  c)=  Xi(w)  we  formed  in  Xi(w)  a  function  that  van 
ished  for  u  =  0  and  congruent  points.     It  is  seen  that 


We  next  wrote  (Art.  96) 


and  we  saw  (Art.  98)  that  every  one-valued  doubly  periodic  function  F(u) 
with  periods  a  and  6  and  which  had  everywhere  in  the  finite  portion  of 
the  plane  the  character  of  an  integral  or  (fractional)  rational  function 
could  be  expressed  in  the  form 

F(u)=  C  + 

+        Z0"(u-  uk)  ----   ±  -Zo*"1'  (u  - 


where  k  extends  over  the  n  infinities  uk  of  F(u)  that  are  situated  within  a 
period-parallelogram,  the  order  of  these  infinities  being  Xk  respectively; 

C  is  an  arbitrary  constant,  while  bk,v  is  the  coefficient  of  -  —in  the 

(u-  uky 

expansion  of  F(u)  in  the  neighborhood  of  the  infinities  u  =  uk  (k  =  I,  2,  .  .  .  n). 
If  r  is  the  order  of  the  function  F(u)  (see  Art.  92),  then  r  =  AI  +  X2  +  -^3 

+     •    •    •     +   Jn- 

The  function  ZQ(U)  is  infinite  of  the  first  order  for  u  =  0.     We  may  next 
write  ;u2+juM 


where  X,  p  are  constants. 
It  follows  that 


i(w) 
u  iu 

We  therefore  have 

ZQ(U)+  2XU  +  fJL  =  Zi(w), 

Zo/(ti)+2A-Z1'(ti)J 

Z0/r(w)=  Zi"(tt),  etc. 
The  formula  above  becomes 

k=n 

F(u)=  C  +5)  K,i  lzi(w  -  u*>  ~  2  ^(w  ~  u$  ~ 
fc=iL 

-  6-       Z'i  -  i*    -  2  A4        Zu  -  Mfc 


DOUBLY  PERIODIC   FUNCTIONS   OF   ANY   ORDER.         433 

k=n  k=n 

The  constants  ^  bk,i(2  Xuk  —  //)  and  ^fbk,22  X 

k=l  k  =  l 

may  be  embodied  in  the  constant  C,  making,  say,  C\.     We  also  note  that 


It  follows  *  that 

t- 
•Ci.4 


1! 

•   ± 


i)  • 

ART.  374.     To  introduce  the  Jacobi  Theory  write 
a  =  2K     and     b  =  2iKf. 

It  follows  at  once  that 


and 


If  we  make  A  =  0,  //  =  -^-,  we  have  from  above 
2  /£ 


and  also 


On  the  other  hand  we  had 


H(u). 

We  may  therefore  write  in  the  formulas  above  ATL(u)  instead  of 

where  A  is  an  arbitrary  constant,  and  Zi(u)=       ^  • 

H(t») 

It  follows  that  we  may  express  every  doubly  periodic  function  F(u)  with 

the  characteristics  required  above  through  the  function        u^  » 

M  (  w) 

*  See  Hermite,  Ann.  de  Toulouse,  t.  II  (1888),  pp.  1-12. 


434  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  375.     To  introduce  the  theory  of  Weierstrass  write 
a  =  2  w     and     b  =  2  at', 

so  that  Xi  (u  +  2  aj)  =  Xx  (w) 

and  « 


We  shall  so  choose  the  constants  A,  /*  that  instead  of  the  function 
we  may  employ  the  function  on.     We  have  the  relations 


a(u  +  2  o>)  =  — 
a(w  +  2<o')=  — 


We  further  have 


It  follows  that 


Comparing  this  result  with 

ff(u  +  2 
it  is  seen  that  we  must  write 

^Xco  =  2y     and     4  Xto2  +  2  /*&>  =  2^w  +  m, 

where  id  has  been  added  to  change  the  sign. 
We  have  at  once 

A--2L    and     P--&, 

2aj  2aj 

and  consequently  also 


This  function  satisfies  the  first  of    the   functional  equations  which   au 
satisfies. 

We  have  further 

2,^u+2,^+,rf<    -^u-2 

^(u  +  2w')=-e    '  w      "  e    w 

or,  since  yw'  —  TI'UJ  =  —  , 


we  have  -^(u  +  2w')  =  — 

It  is  thus  proved  that  ^(u)  satisfies  also  the  second  functional  equation 
satisfied  by  au.     We  may  therefore  put 

ty(u)=  Ban, 
where  B  is  an  arbitrary  constant,  and 


au 


DOUBLY   PERIODIC   FUNCTIONS   OF   ANY   ORDER.         435 
ART.  376.     It  is  evident  from  above  that  we  may  write  F(u)  in  the  form  * 


We  here  have  F(w)  expressed  as  a  sum  of  terms  each  of  which  is  a  complete 
derivative. 

This  formula  is  therefore  especially  useful  in  all  applications  of  the 
elliptic  functions  that  involve  integration.  The  constant  Ci  may  be 
determined  if  we  know  the  value  of  F(u)  for  any  value  of  the  argument 
different  from  the  quantities  uk. 

ART.  377.     We  saw  in  Art,  299  that 


2    pit  -  puk 

where  we  assumed  that  uk  is  not  congruent  to  a  period;  otherwise  £uk  and 
$>Uk  would  be  infinite.  We  therefore  first  exclude  in  this  discussion  all  the 
quantities  Uk  which  are  congruent  to  periods  and  attach  a  star  to  the  sum 
mation  sign  to  call  attention  to  this  fact.  We  have  accordingly,  if  we  note 
the  formulas  of  the  preceding  Article, 

* 


v*  Bkw:(u  -  wo  -  v*  Bk<»;u  -  v 

+*  *i  ** 


2  *-*  pu  -  puk 


We  note  that  the  second  summation  on  the  right  is  a  constant.     Two  cases 
are  possible: 

(1)  None  of  the  quantities  Uk  is  congruent  to  a  period;  or 

(2)  Some  of  the  quantities  uk  are  congruent  to  periods. 

In  the  first  case  we  may  remove  the  star  from  the  summations.  We 
then  have  ^u^Bkw  =  0.  It  then  follows  at  once  that  ^  BkM{(u-uk) 
is  rationally  expressed  in  terms  of  pu  and  p'u.  In  the  second  case  only 
one  of  the  quantities  uk  can  be  congruent  to  a  period  and  therefore  also  to 
zero,  since  the  quantities  uk  form  by  hypothesis  a  complete  system  of 
incongruent  infinities.  This  infinity  may  be  transformed  to  the  origin. 

We  must  consequently  add  Bk(1^u  to  2)*5fc(1)C(w  ~  ft*)  that  we  may 
u  —  uk).  But  here  also  it  is  seen  that 


>u  =  o,     since          Bk™  =  0. 


have  V 


Thus  without  exception  it  is  seen  that       Bkw£(u  —  Uk)  is  rationally  express 
ible  through  pu  and  p'u.  k 

*  *  See  Kiepert,  Crelle's  Journ.,  Bd.  76,  pp.  21  et  seq. 


436  THEORY   OF   ELLIPTIC   FUNCTIONS. 

Further,  since  the  derivatives  of  £(u  —  Uk)  are  all  rationally  expressible 
through  <@u  and  @'u,  it  follows  that 


where  R  denotes  a  rational  function  of  its  arguments.     This  theorem  is 
due  to  Liouville  (see  Art.  155). 

Corollary.  —  If  a  doubly  periodic  function  has  the  property  of  being 
infinite  only  at  the  point  u  =  0  and  congruent  points,  then  this  function 
F(u),  say,  is  an  integral  function  of  pu  and  <@'u.  To  prove  this  note  that 
since  u  =  0  is  the  only  infinity  within  the  first  period-parallelogram 
we  have  k  =  1  and  u\  =  0.  Further,  since  2)  5*(1)  =  0,  it  follows  that 
#!(!)=  o.  We  thus  have 


By  definition  we  had  * 

£:«= 

and  consequently 


'u  +  p'upu), 

/r+2^  +  ^V), 
"+  3  >+  3 


It  follows  that  F(w)  is  an  integral  function  of  g?(w)  and  g?'(w). 

ART.  378.     Let  F(u)  be  a  doubly  periodic  function  of  the  second  sort 

so  that 

F(u  +  a)=  vF(w), 

F(u  +  6)  =  i/F(tt). 
The  logarithmic  derivative  of  F(w), 


is  a  doubly  periodic  function  of  the  first  sort.  The  function  (£>(ii),  as  seen 
in  Art.  4,  becomes  infinite  on  the  zeros  and  on  the  infinities  of  F(u).  Let 
^i°?  ^2°,  •  •  •  j  um°  be  the  zeros  of  F(u)',  and  at  m°  let  F(u)  be  zero  of  the 

*  See  Kiepert,  Dissertation  (De  curvis  quarum  arcus,  etc.,  Berlin,  1870). 


DOUBLY   PERIODIC   FUNCTIONS   OF   ANY   ORDER.         437 

/\-  order  (i  =  1,  2,  .  .  .  ,  m).  Let  u\,  u2,  .  .  .  ,  un  be  the  infinities  of 
F(u);  and  at  Uj  let  F(w)  be  infinite  of  the  ///  order  (/  =  1,  2,  .  .  .  ,  ri). 
We  may  therefore  write 

F(u)  =  (u  -  Ul°)«Fi(u)     (i  =  1,  2,  .  .  .  ,  TO), 

where  Ft-(w)  is  neither  zero  nor  infinite  for  w  =  wt°. 
It  follows  that 


and  consequently  Rea^(u)  =  /U; 

and  similarly  Re^(u)  =  -W. 

U=U; 

It  is  thus  seen  that  (f>(u)  has  only  infinities  of  the  first  order.  It  was  seen 
in  the  previous  Article  that  if  the  development  of  ^(M)  in  the  neighbor 
hood  of  its  infinities  is  given,  we  may  express  (j>(u)  through  the  ^-functions. 
It  follows  also  here  that  the  quantities  Bk(v+l)  are  all  zero,  and  conse 
quently 

-  U2°)  +  '    •     '  +^(U  -  Un°) 
u  -  U2)~  '    •    •  -j*mr(u  -  Un). 

Also,  since 


it  is  seen  that 

F'(u) 


j  ' 


Through  integration  it  follows  that 
F(u)  =  ec>u+c>  +(u  ~  MI°)A 


Every  doubly  periodic  function  of  the  second  sort  and  consequently  also 
every  doubly  periodic  function  of  the  first  sort  may  be  expressed  in  this 
manner.  This  representation  corresponds  to  the  decomposition  of  a 
rational  function  into  its  linear  factors  (see  Arts.  12  and  26).  Instead 
of  the  function  ty(u)  we  may  write  either  H(M)  or  au. 

Further,  since  the  sum  of  the  residues  of  a  doubly  periodic  function  of 
the  first  sort  (Art.  99)  is  zero,  we  have 

2Res</>(«)=  Sx-Sju  =  0, 
or  2/i  =  SJM  =  r, 

where  r  is  the  order  of  the  function  F(u).  It  follows  also  that  a  doubly 
periodic  function  of  the  second  sort  F(u)  has  as  many  zeros  of  the  first 
order  as  it  has  infinities  of  the  first  order,  a  zero  or  infinity  of  the  yth 
order  counting  as  v  zeros  or  infinities  of  the  first  order. 


438  THEORY   OF   ELLIPTIC    FUNCTIONS. 

ART.  379.     We  may  write 


(A)  F(u)  =  ecuc'         ~  ~    2      -   -   -   a(u- 


a(u  —  u\)o(u  —  U2)   •    •    -    o(u  —  ur) 

where  some  of  the  quantities  UIQ,  u2°,  .  .  .  ,  ur°  may  be  equal  and  some  of 
the  quantities  HI,  u2,  .  .  .  ,  ur  may  be  equal.  This  representation  of  a 
doubly  periodic  function  is  very  convenient  when  all  the  zeros  and  infinities 
are  known. 

We  have  assumed  that  the  points  m°  and  Uj  all  lie  within  the  same 
period-parallelogram.  This  assumption,  however,  is  not  necessary;  for 
if  2  co  be  added  to  or  subtracted  from  the  argument  of  one  of  the  ^-functions 
which  enters  in  the  expression  above,  then  only  the  factor  before  the  frac 
tion  is  changed. 

For  example. 

a(u-  up-2u))=-  e-2*(t*-tt,-o,)  0(u  _  Up)f 

or  o(u  —  Up)  =  -  e^(u-Up-aj)  0^u  _  ^+  2  CD)]. 

It  follows  that  every  elliptic  function  of  the  rth  degree  may  be  expressed 
in  the  above  form  in  an  infinite  number  of  ways. 

ART.  380.  It  we  write  u  +  2  a>  in  the  place  of  u,  then  a(u  —  ur)  be 
comes—  ^(ti-ii'+w)  o(u  —  uf),  where  U'  =  UIQ,  u2°,  .  .  .  ,ur°;ui,u2,  •  .  .  ,  ur. 
Hence,  since  F(u  +  2  w)=F(u)  [if  we  suppose  that  F(u)  is  a  doubly  periodic 
function  of  the  first  sort],  it  follows  that 


(B)      F(u)  =  i(*+?+<  6     ^  a(u~  Ul°)a(u  ~  U2°}  '  '  '  °(U  ~  Ur°}  - 

-2^2% 
e      i=1  a(u  —  Ui)o(u  —  u2)   •   -   •   o(u  —  ur) 

The  two  expressions  (A)  and  (B)  must  be  equal.     We  must  consequently 
have 


or  e 

/t=r         i=r      \ 

and  similarly  e  i=l     i==1       —  1. 

In  virtue  of  these  relations  we  also  have 

(1)  2  co,  +  2  T?  (j^Ui  -  j?uA  =  2  Mm, 

/i  =  r  i  =  r        \ 

(2)  2  co)f  +  2  ^  (  2)^i  -  2)  w.'°  )  =  2  M'o, 

where   M   and  M'   are   integers    (positive   or   negative,  including   zero). 


DOUBLY   PERIODIC    FUNCTIONS    OF   ANY   ORDER.         439 

From  the  two  equations  just  written  it  follows  that 

But  since  yaj'  —  o»/  =  ^  xi,  it  is  seen  that 

c  =  2  M'i)  -  2  A/y. 
If  c  is  eliminated  from  (1)  and  (2),  we  have 


,MI  -  2ju.-°  r  2  M<»'  ~  2  M'w. 

t  =  i        1  =  1 

For  the  sake  of  greater  simplicity  we  may  write  —  m!  for  M  and  m  for  Af '. 
We  then  have 

c  =  2  mTj  -}-  2  m  T)  f 


where  m,  m'  are  positive  or  negative  integers  or  zero.  This  theorem  is 
due  to  Liouville.* 

From  the  latter  relation  it  is  seen  that  if  the  r  infinities  of  a  doubly 
periodic  function  of  the  rth  order  have  been  chosen,  then  only  r  -  1  of 
the  zeros  are  arbitrary. 

As  we  saw  above,  we  may  write  for  a  zero  another  zero  that  is  con 
gruent  to  it.  We  may  therefore  increase  ur°  by  ur°  +  2  ma>  +  2  m'aj'. 
If  this  is  done,  then  for  the  new  system  of  zeros  and  infinities  we  have 
m  =  0  =  m'  and  consequently 


n     and     c  =  0. 

»=i         1=1 
We  then  have 

F(u)  =  C  °(u  ~  u^°(u  ~  ^2°)  '   •   •  o(u  —  ur°)  ^ 
o(u  —  ui}a(u  —  u2)  •  •  •  o(u  —  ur) 

It  is  thus  seen  that  F(u)  depends  upon  the  quantities  2  w,  2  a>'  ',  C; 
u\j  U2,  .  .  .  ,  ur\  and  upon  r  —  1  of  the  quantities  uf  (we  note  in  partic 
ular  that  of  the  r  quantities  uf  there  are  only  r  —  I  arbitrary).  It- 
follows  that  the  function  F(u)  depends  upon  2  r  +  2  constants.! 

*  Liouville  (Lectures  delivered  in  1847,  published  by  Borchardt,  Crelle,  Bd.  88,  p.  277, 
or  Liouville,  Comptes  Rendus,  t.  32,  p.  450)  proves  this  important  theorem  and  also 
the  two  fundamental  theorems  already  given,  viz.:  a  doubly  periodic  function  of  the 
nth  order  may  be  expressed  rationally  through  an  elliptic  function  of  the  second  order 
and  its  derivative;  a  doubly  periodic  function  must  become  infinite  at  least  twice  within  a 
period-parallelogram.  Prof.  Osgood,  Lehrbuch  der  Funktiontheorie,  p.  412,  uses  these 
three  theorems  as  the  foundation  of  his  treatment  of  the  doubly  periodic  functions. 

t  See  Schwarz,  loc.  cit.,  p.  20,  or  Kiepert,  Crelle,  Bd.  76,  p.  21;  or  Appell  et  Lacour, 
Fonct.  Ellip.,  p.  48. 


440  THEORY   OF   ELLIPTIC   FUNCTIONS. 

The  expansion  of  the  function  F(u)  through  H(^)  in  the  place  of  au 
may  be  derived  in  a  similar  manner  (see  Riemann-Stahl,  Elliptische  Func- 
tionen,  p.  110). 

Corollary  I.  —  We  note  that  the  function  F(—  u)  is  an  elliptic  function 
of  the  same  nature  as  the  function  F(u)  considered  above.  It  is  also 
evident  that 

%[F(u)  +  F(—  u)]  =  ^o(u),  say,  is  an  even  function,  and  that 
%[F(u)—  F(—  u}]  =  ^i(u)  is  an  odd  elliptic  function. 

That  every  elliptic  function  my  be  expressed  as  a  sum  of  an  even  and  an 
odd  elliptic  function  is  seen  from  the  identity 

F(u)=  l[F(u)+F(-  u)]  +  l[F(u)-  F(-  u)], 
or  F(u)=  ir0(u)+^i(u). 

Corollary  II.  —  We  may  next  prove  that  every  even  elliptic  function  of 
order  say  2  r  may  be  rationally  expressed  through  gw.  Such  a  function 
may  be  represented  in  the  form 


We  may  also  write 

a(u  -  Ui°)a(u  +  uf) 

o(u  -  Ui°}a(u  +  UJQ)  __  a2uo2Ui° 


0(U  —  Ui)(7(u  +  Ui)  0(U  —  Uj)o(u  +  Uj) 


02U02Ui 


(J2Uj° 


We  therefore  have 


4=1  1=1 

a  formula  by  which  it  is  shown  that  -^oO)  is  rationally  expressed  through 
We  may  therefore  write 


where  RQ  denotes  a  rational  function  of  its  argument.     Further,  if 

is  an  odd  elliptic  function,  then,  since  <@'u  is  also  an  odd  elliptic  function, 


is  an  even  elliptic  function  =  RI($>U),  say, 

so  that  ty\(u)=  8>'u  R  i  (&u) , 

where  RI  denotes  a  rational  function  of  its  argument. 


DOUBLY   PERIODIC   FUNCTIONS   OF   ANY   ORDER. 


441 


ART.  381.     As  an  interesting  application  of  the  above  representation 
of  an  elliptic  function  we  note  the  following: 
In  determinant al  form  we  write  the  formula 


G2UG2V  1,    %>V 

We  may  also  express  through  sigma-quotients  such  expressions  as 
1,     pu,     p'u 
1,     pv,     $>'v    =  A(w),  say. 

The  infinities  of  pu  and  p'u  are  congruent  to  the  origin,  $>u  being  infinite 
of  the  second  and  p'u  of  the  third  order  for  u  =  0.  The  determinant  is  a 
doubly  periodic  function  of  the  third  order  in  u  with  zeros  HI°=  v,  u2°  =  w 
and  u3°=  -  v  —  w.  Further,  Ui°  +  u2°  +  u3°=  0  =  2  (infinities),  the  infin 
ities  being  the  triple  pole  zero. 

It  follows  then  that  the  determinant  must  be  of  the  form  * 

r  a(u  +  v  +  w)a(u  —  v)a(u  —  w)a(v  —  ir)  _     < 

v-'  „ ~ ..  —    —1  Ut 


Multiply  both  sides  of  this  expression  by  u3  and  then  make  u  =  0,  and  we 
have 

p  g(y  +  w)0(v  —  w)  _ 
a2va2w 


1, 


so  that  C  =  -  2.     It  follows  that 


o(u  4-  v  +  w)a(u  —  v)a(u  —  w)a(v  —  w)  _       1 


1 ,     tpv,      o* 

1        &W        £>' 

Appell  and  Lacour  (loc.  tit.,  p.  63,  Ex.  2)  give  an  incorrect  value  to  the 
constant  C. 

Further,  since  p'uj  =  0  =  $/o/,  if  we  write  in  the  expression  above  v=aj 
and  w  =  a/,  it  becomes 


a(u 


-f  (t)')a(u  —  a))a(u  —  ajf}o(co  —  CD') 


or 


and  consequently 


r     i  >\ 

fj(oj  +  (*) ) 


—  a>  — 


o(u 


U  —  CD') 


0*1*0*0) 


=  4  (<pu  —  e2)(<pu  -  e-i)(<?u  -  63). 

*  See  Daniels,  Am.  Journ.  Math.,  Vol.  VI,  p.  266. 


442  THEOEY   OF   ELLIPTIC    FUNCTIONS. 

ART.  382.     The  fourth  method  of  the  representation  of  the  doubly  peri 
odic  functions  is  as  follows: 
We  had  in  Art.  376 

F(u)  =  d  +  5)£*<i>  C(u  -  uk)  +  2)5,(2)  p(u  -  uk) 

k  k 


In  Art.  272  we  saw  that 


and  consequently 


If  we  take  the  summation  over  this  expression  with  regard  to  k  and  note 
that  the  summations  with  regard  to  w  and  with  regard  to  k  may  be  inter 
changed,  we  have 


We  further  note  that 

V  1 


(u  -  U 

(w  -  ^)  =  -  2!  V 

** 


, 

+ 


,\  1  1? 

<  ;  -  rz  --  J? 

f  •(»  -  t*fc-  ^)2      w;2  ) 


(u  -  uk-  w)9 
-  -  l-  -  -,  etc. 


It  follows  at  once  that 


-  Uk 


^  (  (u  -  uk-  wy 

+|^__^_  +  ||_^L 

•*) 


DOUBLY   PERIODIC    FUNCTIONS    OF   ANY    ORDER.         443 
If  for  brevity  we  put 


(u  -  nk)2  (u  - 

the  above  formula  may  be  written 

u)  =  Ct  +  /(u) 

u-      '  A; 

ART.  383.     We  may  next  consider  the  fifth  kind  of  representation  of 
the  doubly  periodic  function  F(u). 
We  saw  in  Art.  287  that 

2ft*-*-* 


ion 

where  22  =  ^  =  e  w  . 
We  have  at  once 

0  +  Z~l  =  t  4-   1 
Z  -  Z~l         t  -   I 

If  we  write 

(M-M*)-  ^  «t^" 

6          "  =  —  ,    where  tk=  e    a  ' 

tk 
it  follows  that 


+ 
Next  let 


and  observe  that  /i  (0  =  0  f or  t  =  0  and  f or  t  =00. 
We  may  then  write  the  formula  for  F(u)  in  the  form 


A-  =  1,  2,  .  .  .  ,  n         \ 
1,2,  ...  ,4-  I/ 
We  have  the  following  expansion  (Art.  286) : 


444  THEORY    OF   ELLIPTIC    FUNCTIONS. 

It  is  further  seen  that 


1 =  --L_  aild          tk        -      tik 


t 

~~\2         (t  -  tk)* 

\fk~  V 

Next  let 


^(2) 


It  is  evident  that       /2(0)  =  0  =  /2  (<*>)• 

The  terms  in  F(u)  which  correspond  to  v  =  1  are 


The  terms  in  F(u)  which  correspond  to  v  =  2  are 


If  we  differentiate  the  formula  above  for  pu  we  have  a  suitable  expres 
sion  for  p'u  in  the  form  of  an  infinite  summation,  which  may  be  written 


where  /3(0  is  a  rational  function  in  t  having  the  property  that 

/3(0)=o=/3M. 

We  continue  this  process  and  finally  write 


the  f  unction  /(O  being  a  rational  function  in  t  such  that 


We  therefore  have 

F(u)=  Ci- 


Since  <  has  the  period  2  w,  it  is  evident  that  F(w)  has  the  period  2w;  also 
noting  that  <  becomes  h2t  when  w  is  increased  by  2  a/,  it  is  seen  that  2  a/ 
is  also  a  period  of  F(u)  provided  the  above  series  is  convergent. 


DOUBLY   PERIODIC    FUNCTIONS   OF   ANY   ORDEK.         445 

ART.  384.  We  may  establish  the  convergence  of  the  series  in  the  pre 
vious  Article  as  follows:  Since  f(t)  =  0  for  t  =  0,  we  observe  that  t  =  0 
is  a  root  oif(t)  =  0,  so  that  we  may  write 


•   •  -f  6^ 
It  is  always  possible  to  choose  t  so  small  that 

|W|  +  |  MM  +  •  •  •  +  IV"  I  <  i- 

It  follows  that*  the  denominator  in  the  fraction  above  is  greater  than  J, 
while  the  numerator  is  finite.     We  may  therefore  write 

f(t)<At, 

where  A  is  a  finite  quantity.     It  is  further  seen  that 

f(h*t)  <  Ah*t, 
f(h*t)  <  Ah*t, 

It  follows  that  the  series  f(t)  +  f(h2t)  +  f(h4t)  +  •  -  -  is  convergent;  and 
in  the  same  way  it  may  be  shown  that  f(h  ~2  t}  +  f(h  ~4  0  4-  •  •  •  is  conver 
gent.  We  have  therefore  established  the  convergence  of  the  series  express 
ing  F(u). 

ART.  385.  We  may  also  express  F(u)  in  the  form  of  an  infinite  product 
whose  factors  are  rational  functions  of  t. 

In  Art.  380  we  derived  the  formula 


a(u  — 

where  MI°  4-  u  2°+  •   •   •  +  ur°=  HI+  u2+ 
In  Art.  291  we  saw  that 


If  for  brevity  we  write 


it  follows  that 

t          -,  i    _  ^2n(^  i    _  /,2n_L 

2-      2  «  <7  "  ' 


with  corresponding  formulas  for  a(u  —  ilk). 


446  THEORY   OF   ELLIPTIC    FUNCTIONS. 

We  next  write 


n 


and  note  that  /i  (0)=  1  =/I(QO). 
We  have  at  once 

^- 

F(u)=Ce  " 


That  the  product  on  the  right-hand  side  is  absolutely  convergent  may  be 
proved  by  writing 


where  /0(0  =  0  =/o(°°);  it  then  follows  by  Art.  17  that  the  above  product 
is  absolutely  convergent  if 

m  =  +oo 


is  absolutely  convergent.     The  convergence  of  this  series  is  easily  estab 
lished  by  using  a  geometric  progression  whose  ratio  is  h2. 

ART.  386.  We  saw  in  Art.  377  that  every  one- valued  doubly  periodic 
function  which  has  everywhere  in  the  finite  portion  of  the  plane  the  char 
acter  of  an  integral  or  (fractional)  rational  function  may  be  expressed 
rationally  through  <@u  and  p'u,  say 

(f>(u)  =  RI($>U,  %>'u), 
where  R  i  denotes  a  rational  function  of  its  arguments.     It  follows  that 

~\  7~>  *\  ~D 

,,,    \       o/ii     /       ,     o/ii      //. 


Writing  for  <@"u  its  value  p"u  =  6  <^u  —  \  g2,  it  is  seen  that  (/>'(u)  may  be 
rationally  expressed  through  pu,  <@'u.     We  therefore  write 


where  R2  is  a  rational  function  of  its  arguments. 

Any  rational  function  of  pu  and  g/w  may  be  written  in  the  form 

T>  /         /  \      ^i  (v 
R  !  (&u,  ®'u)  =     1|S 


where  G\  and  G2  are  integral  functions. 


DOUBLY   PERIODIC    FUNCTIONS   OF   ANY   ORDER.         447 

Further,  since 

(p'u)2 

it  is  evident  that  we  ma    write 


where  S,  T  and  W  are  integral  functions  of  <@u\  or  finally 


where  U  and  F'are  rational  functions  of  <@u.     We  have  accordingly 

(1)  <f>(u) 
and  similarly 

(2)  6'(u) 


where  U\  and  Vi  are  rational  functions  of  pu.     We  note  that  V  and  V\ 
cannot  be  simultaneously  zero;    for   U(pu)  and   U\(pu)  are   both   even 
functions  of  u,  while  if  <p(u)  is  even  <j>'(u)  must  be  odd  and  vice  versa. 
From  (1)  and  (2)  it  follows  that 


(3)    ?'u  =  and     (4) 


In  general  both  of  these  equations  (and  always  one  of  them)  have  definite 
forms,  since  V  and  V\  cannot  both  be  simultaneously  zero.  If  then  the 
values  <t>(u)  and  <$u  are  known,  then  p'u  is  uniquely  determined. 

If  in  the  equations  (1)  and  (2)  neither  V  nor  Vi  is  zero,  by  eliminating 
p'u,  we  have 

(I)  0{« 


where  g  denotes  an  integral  function  of   its  arguments.     If  further  we 
square  the  equation  (3)  and  give  to  p'u2  its  value  in  terms  of  #ra,  we  have 

(II)  gi{s>u,<t>(u)}  =  0, 

where  g\  is  an  integral  function. 

On  the  other  hand  if  V,  say,  is  zero,  we  have  from  (1)  the  equation 

(I')  g  {  <pu,  $(u)  }  =  0,  and  from  (4) 

(II')  £i{^,<£'00}  =  0, 

where  g  and  g\  are  integral  functions.     We  thus  always  have  two  algebraic 
equations  among  the  three  functions  pu,  <j>(u),  <j>'(u). 

Under  the  assumption  that  the  pair  of  primitive  periods  2  o>,  2  wf  of  pu 
are  at  the  same  time  a  primitive  pair  of  periods  of  </>(u)  it  may  be  shown 
that  the  two  equations  (I)  and  (II)  or  (I7)  and  (II')  have  only  one  common 
root  in  pu. 


448  THEOKY   OF   ELLIPTIC    FUNCTIONS. 

The  following  indirect  proof  is  due,  I  believe,  to  Weierstrass:  Suppose 
that  a  pair  of  values  belonging  to  <fr(u)  and  <j>'(u)  has  been  chosen  and 
suppose  that  the  equations  (I)  and  (II)  have  two  common  roots,  say 

<@u  =  Si     and     <@u  =  82. 

Suppose  that  u\  is  the  value  of  u  which  satisfies  the  equation 

$>ui=  si. 

Then  also,  since  <@u  is  an  even  function,  the  value  —u\  satisfies  the  same 
equation. 

From  the  equation  (3)  above  we  have 


The  two  values  that  are  had  through  the  extraction  of  the  root  are  +@'u 
and  —<@'u  and  there  is  only  a  choice  of  u  between  +u\  and  —u\.  We 
shall  suppose  that  +u\  gives 


By  a  comparison  of  (a)  and  (b)  it  is  seen  that 

<l>(u)=  <f>(ui). 
Next  suppose  that  u2  is  the  value  of  u  which  satisfies  the  equation 


then  also  —  u2  satisfies  the  same  equation. 

In  the  same  way  as  the  equations  (a)  and  (b)  were  formed,  we  have 


and 

„'„ 


V(s2) 


It  follows  that  </>(u)=  <j>(u2),  and  consequently  corresponding  to  <j>(u)  to 
which  a  definite  value  was  given  at  the  outset,  we  have  shown  that 


In  the  same  way  from  the  value  of  <f>'(u)  which  was  chosen  at  the  outset 

we  have 

(ii) 


DOUBLY   PERIODIC   FUNCTIONS   OF   ANY   ORDER. 


449 


In  Art.  37a  it  was  seen  that  if  the  relations  (i)  and  (ii)  are  true,  then 
u\  —  u2  is  a  period  of  <t>(u).  It  follows  that,  if  2co  and  2w'  are  a  pair  of 
primitive  periods  of  this  function, 

u\  —  u2  =  2  mw  +  2  m'a)', 

where  m  and  mf  are  integers.  We  have  thus  shown  that  the  two  equations 
(I)  and  (II)  have  only  one  common  root.  The  method  to  be  followed  is  the 
same  if  we  take  the  equations  (F)  and  (!!')• 

It  may  be  shown  *  that  if  two  algebraic  equations  have  only  one  root  in 
common,  then  this  root  may  be  expressed  rationally  in  terms  of  the  coeffi 
cients  of  the  two  equations,  so  that  therefore  here 


where  R$  is  a  rational  function  of  its  arguments.  In  this  connection  note 
the  proof  due  to  Briot  and  Bouquet  in  Art.  156. 

It  follows  then  as  was  shown  in  Art.  158  that  every  transcendental  one- 
valued  analytic  function  which  has  an  algebraic  addition-theorem  is  necessar 
ily  a  simply  or  a  doubly  periodic  function. 

ART.  387.     It  follows  from  Art.  376  that 


C0+ 


0(U  -  W*)- 


M  -  uk)  =  #(u  -  U 
Since  2)  ^(1)  =  0,  we  may  write 
V  Bk™  log  <J(M  -  uk)  =  V 

~~  — 

We  also  saw  in  Art.  299  that 

-  5)  BkWr(u  -  Uk)  =-'u 
k 

It  follows  that 


=  2,  3,  .  .  .  ,  lk  -  1]. 


log  o(Uk  ~  M)  +  Constant. 


an  cZZtpfic  function  of  u. 


where  ^>i(w)  is  a  doubly  periodic  function  with  periods  2cu,  2w'.     Further, 
since  (Art,  356) 

log 


ou  auk 
*  See  Baltzer,  Theorie  der  Determinanten,  p.  109. 


450 
we  have 


THEORY    OF   ELLIPTIC    FUNCTIONS. 


j*F( 


u)du  -  u 


The  moduli  of  periodicity  of  the  general  integral  /  F(u}du  are  therefore 

had  at  once;  at  the  same  time  it  is  seen  that  this  general  integral  may  be 
expressed  through  (see  also  Chapter  VIII): 

1.  An  elliptic  integral  of  the  first  kind; 

2.  An  elliptic  integral  of  the  second  kind; 

3.  A  finite  number  of  elliptic  integrals  of  the  third  kind; 

4.  A  rational  function  of  pu  and  p'u. 


EXAMPLES 

1.   Show  that  any  integral  function  of  pu  and  p'u  may  be  written  in  the 
form 


<J(M)W 

where  uf,  u2°}  .   .   .   ,   un°  are  the  zeros  of  the  function. 

2.   Show  that  any  rational  function  of  pu  and  p'u  may  be  written 

-  a^'u  +  -   •   -  +  an&fr-Vu 


F(u)  =  A  a° 


60  + 


+ 


3.   Write 


•>(n-i). 


(n'li  c->(n~^u 

Vuu  •   •   •  i     5^          «i 


u} 


-u)  ...  a(un  —u)a(u 


Show  that 


where  C  is  independent  of  u. 

Multiply  both  sides  of  this  expression  by  un+l  and  determine  C. 

4.  Express  F(u)  through  the  function  Z^w)  of  Art.  374,  and  derive  the  expres 
sion  corresponding  to  the  one  of  Art.  387  for  the  integral  /  F(u)du  in  terms  of 
Z(w)  and  the  theta-f  unctions. 


CHAPTER  XXI 


THE  DETERMINATION  OF  ALL  ANALYTIC  FUNCTIONS  WHICH 
HAVE  ALGEBRAIC  ADDITION-THEOREMS 

ARTICLE  388.  The  problem  of  this  Chapter  has  already  been  solved 
for  the  case  of  the  one-valued  functions.  Weierstrass  *  has  also  solved  it 
for  the  many-valued  functions  by  making  use  of  the  principles  which  we 
shall  attempt  to  give  in  the  sequel.  Using  a  method  due  to  him  (see 
references  in  Chapter  II)  we  must  first  show  that  a  function  <j>(u)  which 
has  an  algebraic  addition-theorem  may  be  extended  by  analytic  continu 
ation  over  an  arbitrarily  large  portion  of  the  plane  without  ceasing  to  have 
the  character  of  an  algebraic  function;  that  is,  in  the  neighborhood  of  any 
given  point  the  function  may  be  developed  in  a  convergent  series  accord 
ing  to  powers  of  a  certain  quantity  which  may  stand  under  a  root-sign, 
and  in  which  series  the  number  of  negative  exponents  is  finite.  We 
assume  that  the  function  may  be  defined  in  the  neighborhood  of  a  certain 
region  about  the  origin  and  we  choose  a  point  UQ  such  that  one  branch  of 
the  function  <j)(u)  has  the  character  of  an  integral  function  at  the  point  UQ. 

We  may  therefore  write 

(1)"  <J)(U)= 

Next  put  «  =  «„+«', 

v  =  UQ  +  v',     UQ  being  a  constant. 

Since  <p(u)  has  by  hypothesis  an  algebraic  adoption-theorem,  we  have  an 
equation  of  the  form 


where  G  denotes  an  integral  function  of  its  arguments. 
We  therefore  have 


O,  <£(«o  +  V),  <t>(2uQ+  u'+  v')}=  0. 
Further,  if  we  write 

U   =   UQ, 

V  =   UQ  +  Uf  +  V', 

it  is  seen  that 

G{<£(MO),  <t>(u0+  u'+  v'),  (t>(2u0+  u'+  v'}}=  0. 

*  See  Forsyth,  Theory  of  Functions,  Chap.  XIII;  or  Phragmen,  Acta  Math.,  Bd.  7, 
p.  33;  I  wish  to  mention  in  particular  the  Berlin  lectures  of  Prof.  H.  A.  Schwarz, 
which  have  been  used  freefy  in  the  preparation  of  this  Chapter. 

451 


452  THEORY   OF   ELLIPTIC   FUNCTIONS. 

If  (j>(2  UQ  +  u'  +  v')  be  eliminated  from  these  two  equations,  there  results 
an  algebraic  equation  of  the  form 


u 


'  +  v')}=  0. 


We  may  consider  <J>(UQ)  as  a  new  constant. 
Writing 

(J)(U0+  U')=  </>l( 


we  see  that 


If  in  equation  (1)  we  write  WQ+  M    instead  of  u,  we  have 


from  which  it  follows  that  by  a  change  of  the  origin  the  function  <j>(u) 
may  be  changed  into  the  function  (f>i(uf)  in  such  a  way  that  the  function 
(f>\(uf)  has  the  character  of  an  integral  function  at  the  point  u'=  0  in  the 
branch  of  the  function  under  consideration. 

Hence  without  limiting  the  generality  of  the  given  function  <p(u)t  we 
may  assume  that  the  point  u  =  0  in  the  branch  in  question  of  the  function 
<j)(u)  is  a  point  at  which  </>(u)  has  the  character  of  an  integral  function. 
Making  this  assumption  suppose  next  that  p  is  the  radius  of  the  circle 
of  convergence  of  the  series  expressing  </>(u)  in  the  neighborhood  of  u  =  0. 

If  then  |  u  |  <  p,  the  function  <j>(u)  has  the  character  of  an  integral  func 
tion  in  the  branch  considered. 

If  |  u    <  i  p,     v  I  <  J  fi,  then  is    \  u  +  v  \  <  p,  and  we  have 


for  the  region  considered. 

If  in  this  equation  v  is  put  =u  it  follows  that 


which  is  an  algebraic  equation  between  ^>(u)  and  (f>(2u)  with   constant 
coefficients.     We  may  write  this  equation 


(2)  G1{^(w 

If  in  this  equation  the  value  of  u  is  limited  so  that    u     <  J  p,  then  within 
this  region  <£(2  u)  has  the  character  of  an  integral  function,  since  \2u\  <  p. 


MANY-VALUED   ELLIPTIC   FUNCTIONS.  453 

Suppose  that  for  (/>(u)  its  expression  as  a  power  series  in  terms  of  u  is 
written  in  equation  (2)  which  is  then  solved  with  respect  to  (f>(2u).  We 
know  that  one  root  of  this  equation  represents  the  branch  of  <p(2  u}  under 
consideration  if  |  u  \  <  J  p.  But  the  coefficients  of  this  equation  may  be 
analytically  continued  throughout  the  whole  region  of  the  circle  with  the 
radius  p.  In  .this  extended  region  with  the  radius  p  the  function  </>(2  u) 
retains  the  character  of  an  algebraic  function.  Hence  the  definition  of  the 
function  may  be  extended  to  a  wider  region  than  the  original  and  indeed 
to  a  region  with  the  radius  2  p. 

By  writing  2  u  for  u  in  the  equation  (2)  we  have 


Eliminate  <j>(2  u)  from  this  equation  and  equation  (2)  and  we  have  an 
algebraic  equation  of  the  form 


If  the  variable  u  be  limited  to  values  such  that  \u\  <  &  then  by  repeating 

the  above  process  it  is  seen  that  the  function  may  be  continued  to  the 
region  of  a  circle  with  radius  4  p. 

By  repetition  of  this  process  we  come  finally  to  an  algebraic  equation 


from  which  it  is  seen  that  the  original  functional  element  may  be  con 
tinued  over  an  arbitrarily  large  portion  of  the  plane  without  the  function 
(j>(u)  ceasing  to  have  the  character  of  an  algebraic  function. 

It  is  also  easily  shown  that  by  this  continuation  of  the  function  the 
addition-theorem  is  true  for  the  extended  region  (see  Art.  51)  and  that 
all  the  properties  originally  ascribed  to  the  function  remain  true  through 
out  the  analytical  continuation. 

ART.  389.  Suppose  that  the  equation  which  expresses  the  addition- 
theorem 


is  developed  in  powers  of  (j>(u  +  v).     It  takes,  say,  the  form 
(3)     ^«+f)+Pi.i[*(«),^ 


where  the  P's  are  rational  functions  of  <j>(u),  (j>(v). 
In  this  equation  write 

u  +  &i  for  u 

and  v  —  k  L  for  v, 

where  ki  is  a  variable  quantity  which  may  be  limited  to  small  values. 


454  THEORY    OF   ELLIPTIC   FUNCTIONS. 

By  this  substitution  u  +  v  remains  unchanged,  and  the  above  equation 
becomes 

(4)     <j>m*(u  +  v)  +  Pi,i[0(tt  +  ki),<l)(v  -  k1)}pm^-l(u  +  v)+  •   -   -  =  0. 

The  equations  (3)  and  (4)  are  algebraic  and  have  at  least  one  root  in  com 
mon,  viz.,  <f>(u  +  v)  which  belongs  to  the  branch  of  the  function  in  question. 

Through  a  finite  number  of  essentially  rational  operations  we  may  by 
Euler's  Method  derive  the  greatest  common  divisor  of  the  two  equations 
and  thus  form  a  new  algebraic  equation  whose  degree  is  less  than  the 
degree  of  either  of  the  original  equations  unless  these  equations  have  all 
roots  in  common.  This  we  suppose  is  not  the  case. 

Let  the  form  of  the  new  equation  be 

2,i[<£(w),  </>(u  +  &i),  $(v),  (j>(v  -  ki)]^~l(u  +  v) 


where  ra2  <  m\. 

We  write  in  the  above  equation 

u  +  k2  instead  of  u 

and  v  —  k2  instead  of  v. 

That  equation  then  becomes 

(j)m*(u  +  v)  +  P2,i[(j)(u  +  k2),  <t>(u  +  k!+  fc2),  <j>(v  -  k2)} 
$(v  -  kl-  k2}]<]>m*-l(u  +  v)+  -   •   •  =  0. 

It  may  happen  that  for  every  value  k2  this  equation  has  all  its  roots  the 
same  as  those  of  the  previous  equation,  and  consequently  its  coefficients 
do  not  depend  upon  k%.  If  this  is  not  the  case  the  two  equations  have  a 
common  divisor,  and  when  we  derive  this  divisor  we  have  a  new  equation 
of  the  form 


0«,(u 

9  (u 

+  •  •  •  =o, 

where  m^  <  m2. 

This  process  may  be  continued.  Each  following  rrik  is  less  than  the  pre 
ceding.  Finally  we  must  either  have  ra^  =  1,  or  the  two  equations  through 
which  a  further  reduction  is  made  possible  have  all  their  roots  common. 

We  thus  derive  an  equation  of  the  form 


(  v  -  k2),  .  .  .  ,  <t>(v  -  kr},  <j>(v  -  k1-k2), 


.  .  +P,  [same  arguments]  =  0, 

-kr)  J 

the  P's  being  rational  functions  of  their  arguments.     We  may  assume  that 
the  degree  of  this  equation  cannot  be  decreased  by  the  above  process.     It 


MANY-VALUED    ELLIPTIC    FUNCTIONS. 


455 


follows  that  all  the  coefficients  of  the  equation  remain  unaltered  when 
u  is  increased  by  a  certain  quantity  k,  and  v  diminished  by  the  same  quan 
tity  k.  Some  of  the  coefficients  of  the  above  equation  may  be  constants, 
but  they  cannot  all  be  constant,  for  in  that  case  (/>(u  +  v)  would  be  a  con 
stant. 

Suppose  that  Pv  is  one  of  the  variable  coefficients,  which  is  therefore  a 
function  of  both  u  and  v. 

We  may  write  Pv=f(u,v), 

and  will  show  that  Pv  is  a  function  of  u  +  v. 

We  know  that  Pv  =  f(u,  v}  has  the  property  that 

f(u  +  k,v-k)  =  f(u,v). 
We  may  choose  k  so  small  that 


or 


du       dv 


It  follows  that /is  a  function  of  u  +  v. 

We  shall  put  f(u  +  v)  =  ^v(u  +  v)  and  shall  show  that  ^  is  a  one- 
valued  function,  while  </)(u}  may  be  an  arbitrarily  many-valued  function. 

Draw  a  circle  about  u  =  0  with  a  radius  R,  where  R  may  be  taken  as 
large  as  we  wish.  If  wre  then  succeed  in  showing  that  tyv  (u  +  v)  is  one- 
valued  within  this  circle  with  radius  R,  the  theorem  may  be  considered 
proved,  since  R  may  be  taken  arbitrarily  large.  We  knowT  that  in  the 
neighborhood  of  u  =  0,  the  function  (£>(u)  has  the  character  of  an  integral 
function.  We  shall  seek  to  cut  out  of  the  circle  two 
narrow  strips  that  are  perpendicular  to  each  other 
and  which  have  the  property  that  for  all  points 
within  this  cross  the  branch  of  the  function  <j>(u) 
under  consideration  has  everywhere  the  character 
of  an  integral  function.  This  may  be  done  as  fol 
lows:  We  suppose  that  all  the  branch-points  of  <f>(u), 
or  of  the  analytic  continuation  of  the  branch  of 
<l>(u)  under  consideration,  are  known.  This  number 
of  branch-points  is  finite,  since  the  circle  is  finite 
and  the  function  has  the  character  of  an  integral  function.  A  straight  line 
is  drawn  connecting  each  of  these  points  with  the  origin,  and  at  the  origin 
a  straight  line  is  drawrn  perpendicular  to  each  of  these  lines.  We  next 
choose  a  direction  from  the  origin  which  coincides  with  none  of  these  lines 
or  with  the  perpendiculars  to  them.  The  perpendicular  to  this  direction 
through  the  origin  does  not  coincide  with  any  of  the  straight  lines  or  the 
perpendiculars  to  them. 


Fig.  76. 


456  THEORY    OF   ELLIPTIC   FUNCTIONS. 

We  thus  have  two  straight  lines  perpendicular  to  each  other  through  the 
origin  which  within  the  circle  pass  through  no  branch-point  of  the  function 


Through  all  the  branch-points  which  lie  within  the  circle  we  draw 
parallels  to  the  two  lines,  and  among  all  these  parallels  we  choose  those 
which  lie  nearest  the  two  lines.  The  two  pairs  of  parallel  lines  which 
have  thus  been  chosen  form  a  cross-shaped  figure  within  which  no  branch 
point  is  situated,  excepting  always  the  origin,  which  in  the  leaf  under  con 
sideration  of  the  function  is  not  a  branch-point.  The  functions  (j>(u)  and 
<f)(v)  are  one-valued  along  the  middle  lines  of  the  strips  which  form  the 
cross.  We  shall  now  take  the  k's  defined  above  so  small  that  |  ki  \  + 
|  k2  |  +  •  •  •  +  |  kr  |  is  less  than  half  the  width  of  the  more  narrow  of  the 
two  strips.  Then  if  u  moves  along  the  middle  line  of  one  of  the  strips, 
while  v  moves  along  the  middle  line  of  the  other,  all  the  arguments 
which  have  been  used  in  the  formation  of  Pv  are  situated  within  the 
cross.  If  u  and  v  are  added  geometrically,  it  is  seen  that  Pv  =  f(u,  v)  = 
"fyv(u  +  v)  is  a  one-valued  function  for  all  values  of  u  +  v  within  the 
square  that  circumscribes  the  circle  with  radius  R.  It  follows,  since  R  is 
arbitrarily  large,  that  fa  is  a  one-valued  analytic  function  of  its  arguments. 

ART.  390.     If  we  write  v  =  0,  then  fa(u  +  v)  becomes 

4,t  \_  P  \<t>(u)><t>(u  +  *i)»  •  '  •  ,*("  +  *!  +  '   '  '  +& 
"(-*i),  .  .  .  ,<£(-&!-  -  -  -  -kr) 


From  this  it  maybe  shown  as  follows  that  <p(u)  and  fa(u)  are  connected 
by  an  algebraic  equation: 

The  function  fa(u)  is  expressed  rationally  through  <t>(u),<j>(u  +  ki),  .  .  .  , 
<p(u  +  ki+  k2+  -  •  •  +  kr).  By  means  of  the  addition-theorem  <j>(u  +  k  i) 
may  be  expressed  algebraically  through  <f>(u)  and  <£(&i),  and  similarly 
</>(u  +  k2),  etc. 

We  thus  have  an  algebraic  equation  of  the  form 

(5)  H(<f>(u),  +v(u)}  =  0. 

From  the  four  algebraic  equations 

,  0(v),  cf>(u  4-  v)]  =  0, 
,  +v(u)]  =  0, 


H[<f>(u  +  v),  tyv(u  +  v)]  =  0, 

we  may  eliminate  <f>(u),  <j>(v),  </>(u  +  v)  and  have  the  algebraic  equation 

v     =  0. 


MANY-VALUED    ELLIPTIC    FUNCTIONS  457 

Further,  if  we  differentiate  equation  (5)  we  have  an  algebraic  equation 

(6)  H^u),   +v(u),  ^f(u),  1r'(u)]  =  0. 
We  also  have  the,eliminant  equation 

(7)  E[<t>(u},<f>'(u)]  =  0.  (i) 


If  from  the  equations  (5),  (6)  and  (7)  we  eliminate  <f>(u)  and  <j>'(u)  we  have 
the  eliminant  equation 

=  0.  (ii) 


It  follows  then  that  ^V(M)  has  an  algebraic  addition-theorem. 

Since  the  algebraic  equation  (5)  exists  connecting  <f>(u)  and  "fyv(u),  it 
follows  that  <j>(u)  is  an  algebraic  function  of  ^v(u).  We  have  thus  solved 
the  problem  of  determining  the  function  <f>(u)  in  its  greatest  generality. 
The  function  <p(u)  is  the  root  of  an  algebraic  equation,  whose  coefficients  are 
rationally  expressed  through  a  one-valued  analytic  function  tyv(u),  which 
function  has  an  algebraic  addition-theorem.  In  the  Weierstrassian  theory 
the  one-valued  analytic  functions  that  have  algebraic  addition-theorems, 
as  shown  in  Chapter  VII,  are  either 

I,  rational  functions  of  u,  or 

«ri 

II,  rational  functions  of  e  w  ,  or 
III,  rational  functions  of  $>u  and 


TABLE    OF   FORMULAS 

(The  formulas  of  Jacobi  and  of  Weierstrass  in  juxtaposition) 

I. 


/b2sin2<£ 

=  amw p.  241, 

z  =  snu,     Vl  -  z2=  cos0  =  cnu,      Vl  -  &2z2  =  dn u.       .  p.  241. 

Vl  —  k2  sin2(f>  =  A0,  u  =  F(k,  z)=  F(k,  (f>).  .     .     .  p.  285. 


am  0  =  0,  snO  =  0,  en  0  =  1,  dnO  =  l.    .     p.  245 

am(-  u)=-  amu,     sn(-u)=-snu,     cn(-u)=cnu,    dn(-u}=dnu. 
sn2u  +  cn2u  =  1,  k2sn2u  +  dn2u  =  1.      .     .     p.  247. 


II. 


=^     or  =  dnu.      .         p.  243. 

du  du 


=  (sn/w)2==  ^  ~  ™2u)(l  ~  k2sn2u).     ...     p.  247. 

sn'u  =      cnudnu,      ........     p.  247. 

cn'u  =  —  snudnu, 

dn'u  =  —  k2sn  u  en  u. 

(sriu)2=  (1  -  sn2u)(l  -  k2sn2u),     .....     p.  247. 
(criu)2=  (1  -  cn2w)(l  -  k2+  k2cn2u), 
(driu)2=  (1  -  dn2u)(dn2u  -  1  +  /b2). 

(See  also  No.  LVI). 
458 


TABLE   OF   FORMULAS.  459 


III. 

dt 


p.  215. 


F*  -  V4*3-  g2t  -  g3 
t  =  gm, p.  298. 


p.  325. 
—  e3). 

4t*-g2t-g3=  4(t-  ei)(t-e2)(t-e3).     .     .    pp.  191,  200. 

Cl+  02+  €3=  0, 

e22+  «32)  =  -  i^2, 


=  0.     .    p.  408. 


460  THEORY   OF   ELLIPTIC   FUNCTIONS. 


IV. 


K  = 


-/V= 

jn  vn  — 


dz 


.     p.  212. 


F(k,  nx  +  p)=2nK  +  F  (k,  p). 
a,mK  =  ^,    am2K  =  T:  =  2amK,     am(p  ±  2  nK)  =  am  p  ±  nn.    p.  241. 


V. 

K  =  0,     dnK  =  k' p.  245. 

k'2=  I p.  213. 


VI. 


K"£ 


_dz =   C2 


-k'2z2)         Q  Vl  -k'2s 


.     p.  213. 


r^ 

J0  VZ 


iVZ 

-  z2)(l  -k2z2). 


iK',  ...     p.  289. 


VII. 


sn(K  +  iK')  =    ,     cn(K  +  iKr)  =  -  —  ,     dri(K  +  iK')  =  0.      p.  246. 
k  k 


TABLE    OF    FORMULAS. 


461 


CO 


f 

J*. 


where 


VIII. 


ei, 
0, 


dt 


—  9 2$  —  93       Jei    2  V  (t  —6i)(t  —  €2)(t 

S  -  4t3-  g2t  -  g3. 


aj"=  w  +  at' 


IX. 


X. 


iK'=Vei- 


108(1  - 


=        -^±-  p.215. 


p.  215. 


pp.  93,  384. 


p.  215. 


pa>'=<*3,      .      .     .     p.  216. 
pV=0.       .     pp.315,  355. 


.     .     p.  201, 
.     .    p.  201. 


p.  201, 


462  THEORY    OF   ELLIPTIC   FUNCTIONS. 

XI. 
sn(—  u)  =  —  snu,     ......    p.  245. 

cn(—  u)  =  cnu, 
dn(—  u)  =  dnu. 


sn(u  +  K)=--,   ....  p.  245. 

dnu 


dnu 

k' 


dnu 

sn(u  +  2  K)  =  —  snu, 
cn(u  +  2K)  =  —  cnu, 
dn(u  +  2K)=  dnu. 


sn(u  +  iK')=  —  —  ,      ......     p.  246. 

ksnu 


k  snu 


snu 

sn(u  4-  2  iK')  =  sn  u, 
cn(u  +  2  iK')  =  —  cnu, 
dn(u  +  2  iK')  =  —  dn  w. 

sn(w  +  K  +  iK')  = 


kcnu 

A* 

-  -^—  • 
k  cnu 


dn(u  +  K 


+  2  K  +  2  t'K')  =  -  sn  t*, 
cn(u  +  2  K  +  2  iK')  =  en  tt, 


TABLE   OF  FORMULAS. 


XII. 


pu  - 


±  0>0=  63+ 


-  60(63- 


-63 


XIII. 


-  e3 
(See  also  formulas  LIV.) 


463 


p.  317. 


.     .     pp.  355,  369. 


.     .     .    pp.  216,  298. 
p.  305. 


p.  307 


p.  307. 


464 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


cn(iu, 


isn(u,  k') 
cn(u,  k') 

1 


XIV. 


cn(u,  k') 


i  cn(iu,  k) 


cn(iu, 


cn(iu,  k) 


m(iu  +  K,k)=  — — — ,    .     , 
dn(u,  kf) 

,.         TS  7N          ik'sn(u,  kf) 
cn(iu  +  K,k)  = — — Li 

dn(u,  k) 

if     ,    IT-  7  \      k'cn(u,k') 
dn(iu  +  K,k)=       /  '     J ; 
dn(u,  k') 


p.  247. 


p.  261, 


i  •         -Tsr  T\      —  icnfu,  k'} 
sn(iu  +  iK',  k)  =  —  -  /  '     ', 
K  sn(u,  k) 


dn(iu 


u, 


1 


sn(u,  k') 


XV. 


p.  246. 


Function 

Periods 

sn  u 

4  K  and 

2iK' 

cn  u 

4K  and 

2K  +  2iK' 

dnu 

2K  and 

4iK' 

p.  245. 


Function 

Zeros 

Infinities 

sn  u 

2  m  K  +  2  niK' 

2mK+(2n  +  l)iK' 

cnu 

(2m  +  l)K  +  2  niK' 

a 

dnu 

(2m  +  1)K  +  (2n  +  l)iKf 

<( 

(m,  n  integers  including  zero.) 


TABLE   OF   FORMULAS. 


465 


2iK' 


0 


sn 
en 
dn 


XVI. 


u  +  (0,  1,2,  3)K  +  (0,1,2,3)^' 


K 


2K 


3K 


m(u  +  2mK  +  2m'iK')  =  (-  l)msnu, 
cn(u  +  2mK  +  2m'iK')  =  (-  l)m+m'cnu, 
dn(u  +  2  raK  +  2  rn'iK')  =  (-  l)m'dn  u, 
(m,  m'  integers  including  zero.) 


p.  245 


1 

dn  u 

-  1 

—  dnu 

k  sn  u 
i  dn  u 

k  cnu 

ik' 

ksnu 
—  idnu 

kcnu 

-ik' 

ksnu 
ikcnu 

kcnu 

—  ikk'sn  u 

k  sn  u 
ik  en  u 

kcnu 

—  ikk'sn  u 

ksnu 

k  en  u 

k  sn  u 

k  cnu 

en  u 

—  cnu 

dnu 
k'sn  u 

dn  u 
—  k'sn  u 

dnu 
-k' 

en  u 

dn  u 
-  k' 

dnu 

dn  u 

1 

dnu 

-  1 

—  dnu 

k  snu 

—  i  dn  u 

kcnu 

-  ik' 

ksnu 
idnu 

kcnu 

ik' 

k  snu 

—  ik  en  u 

k  en  u 

ikk'sn  u 

k  snu 

—  ik  en  u 

k  cnu 

ikk'sn  u 

k  sn  u 

kcnu 

k  snu 

kcnu 

cnu 

—  cnu 

dnu 
—  k'sn  u 

dn  u 
k'snu 

dn  u 
k' 

dn  u 

k' 

dn  u 

dn  u 

466 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


XVII. 


[See  p.  368.] 


•M 

0 

1 

i 

I 

i 

Vl  +  k' 

-VF 

0. 

Vl  +k' 
Vk? 

Cn 

dn 

-  1 

Vl  +k' 

-Vk' 

-kf 

Vl  +  Iff 
-Vk' 

_  I 

—  i 

Vk  -  ik' 

1 

Vk  +  ik' 

i 

Vk 

Vk 

Vk 

Vk 

Vk 

-Vl  +  k 

-Vik' 

-iVl  -k 

V  -  ik' 

Vl  +k 

Vk 

Vk 

Vk 

Vk 

Vk 

dn 

-Vl  +  k 

Vk'(k'+ik) 

-Vl  -k 

-Vkf(k'-ik) 

-Vl  +  k 

977 

4-   7* 

I 

1 

I 

T 

i'T 

vT-T 

-iVkf 

k 

-ik' 

Vl  -k' 

-iVkf 

I     VT 

dn 

-ikl 

vT^TP 

-iVkf 

k 
0 

Vl  -k' 

iVv 

T  11 

-ikl 

i 

Vk  +  ikf 

1 

Vk  -  W 

—  i 

Vk 

Vk 

Vk 

Vk 

Vk 

Vl  +  k 

V  —  ik' 

-iVl  -k 

-Vik' 

-Vl  +  k 

Vk 

Vk 

Vk 

Vk 

Vk 

dn 

Vl  +  k 

Vk'(k'-ik) 

Vl  -k 

Vk'(k'  +  ik) 

Vl  +  k 

977 

n 

I 

i 

I 

n 

en 

1 

Vl  +  k' 

Vk> 

o 

Vl  +k' 
-Vk' 

..  1 

dn 

1 

Vl  +kf. 
Vk' 

k' 

Vl  +k' 
Vk' 

1 

u        = 


K 


2K 


*  In  the  table  7  =  lim 


=0  ksnu 


TABLE   OF   FORMULAS. 


467 


XVIII. 


&(^\=el+Ve1-e2Vel-e3,     .    .   ,.    ....     p 


.  369. 


=  -  2(6i-  63)v/6i-62-  2(ei-  62)  Vei-  e3, 


-  63, 


(2")  =  ~ 


-  63  Vei-  63 


+  w=  2i(6i-  63)      62-  63  -  2i(e2-  63)      ei-  e3 


-  =  62-  i     e2  -  63  \6i  -  62 


=  ~  2(6l~  62)       s-  63+2i(e2-  63)  VCl-  62. 

(Halphen,  Fonci.  Ellip.,  Vol.  I,  p.  54.) 


468  THEORY   OF   ELLIPTIC    FUNCTIONS. 


Sin  TtU  =  71 

TO 


u  -  m      m 


P.      20e 


.,  TO  =  +oo 

2    =  v        i 

27ra      ^^oo  (u  — 


m)2 


XX. 

-,£ 

q  =  e      K 


p.  220. 

=  1  +  2  5  cos  2  w  +  2  ^4  cos  4  u  +  2  g9  cos  6  u  +   -   •   •  , 


3      (2TO+1)2    (2 

5     4      e 

m  =  -oo 


H^K  -  w),     .     .     p.  221. 
=  !  ~  2  ?  cos  2  w  +  2  54  cos  4  w  -  2  g9  cos  6  w  +  •  -  •  , 


H  is  an  odd  function;  0,  ©j,  HI  are  even  functions. 


TABLE    OF    FORMULAS. 


469 


U       1  tt» 


XXI. 

w  =  2  pa)  4-  2  JJL'C 

^y-o,±i,±2, 

W    7^    0 


du 


u 


u  —  w      w      w 


a(—u)=—au, 


,    .     .    p.  319. 
.     pp.  318,  324. 


~  W  W 


p.  315. 


.     .     .     .     p.  323. 
u).     .     .     p.  298. 


XXII. 


'—      .     .  p.  324. 

....  p.  323. 

....  p.  323. 

•  ....  p.  323. 


XXIII. 

+  2 aj)  =  ru  +  2  T],     C(u  +  2w')=  £u  +  2  >/.      pp.  303,  338. 
C^>,  V  -  C"'»  ^  =  ^  +  V-    -     •     •     P-  301. 

7?a/-w)/=^,  if  R(-}  is  positive p.  339. 

2  V^/ 


470  THEORY   OF   ELLIPTIC    FUNCTIONS. 


XXIV. 

i(u  +  K)=®(u),  0i(w  +  iK')  =  mi(u),  pp.  222,  223. 


K)=  0i  (w), 
H(tt  +  K)=  HI(M), 


0(w  +  2  mK)  =  0(w),  0(w  +  2  wiK') 

(-  l)wH(w),          H(w  +  2?niK') 

rr/  .«_.' 

.     A          TWTTt. 


TABLE   OF   FORMULAS. 


471 


XXV. 


o(u,  w,  a/ 


(0) 


ooj 


,.     pp.  378,  304. 


Hi(0) 


304, 


377. 


(HftT 


(7W 


. 

0(0) 


0(U 

al(u 


=      e 


XXVI  .......     pp.  340,  380. 


'  =  -    2 


2  wr 


2a>') 

2  0/ 


25)=  (- 

2  5    =      - 


|"2  5  =  2  pa>  +  2  ra>',      p,  r  any  integers  including"} 


472 


THEOKY   OF   ELLIPTIC   FUNCTIONS. 


XXVII p.  224. 


Function 


Zeros 


(2m  +  l)K  +  2niK' 

(2m  +  l)K  +  (2n  + 

2mK  +  (2  n  + 

2  mK  +  2  niK' 


(m,  w  integers  including  zero.) 
XXVIII. 


snu  =  —-= 


dnu 


S(u) 


XXIX. 


XXX. 


Function 

Zeros 

t^oW 

m  +  nr  +  - 
2 

^l(«) 

m  +  WT 

*,(*) 

m  +  i  +  nr 

^3(tt) 

m  +  J  +  nr  + 

T 

2 

(m,  n  integers  including  zero.) 
*  =  ^  =  -  (P-  23°)- 


p.  244. 


p.  229. 


TABLE   OF   FORMULAS. 


473 


XXXI. 


Function 

Zeros 

G\U 

(2m 

+  l)w  +  2  no/ 

G2U 

(2m 

+  l)o>  +(2n  + 

IK 

°zu 

2  mw  +  (2  n  + 

IX 

GU 

2maj  +  2na)' 

(m,  n  integers  including  zero.) 


OU 


GU 


u  =  _ 


p.  380. 


XXXII p.  384. 


v/e-^=£^  =  <^,/.  v-ir^;=2ff=.«^W'; 


G<JJ  OOHJOJ 


GO)          ooj  aw 


€*>  "   (JW   . 


ooj  aw 


aco 


aw  aw  aw 


aa> 


acu 


—  e\  =  — 


-  e3,     Ve2—  ei=  — 


—  e2, 


where  R 

iw 


474  THEORY   OF  ELLIPTIC   FUNCTIONS. 


XXXIII p.  230. 


=  q-le-2rnut 


nr)  =  (- 


m 


nr)  =  (- 


_  /->—  n2/,  —  2n7fiu 


TABLE   OF   FOKMULAS. 


475 


XXXIV. 


p.  386. 


=  ±- 


—  62 ei  — 


=  ±    e2 


Vi 


4/ 4 


[Schwarz,  ^oc.  c^.,  p.  26.] 


476  THEORY   OF   ELLIPTIC    FUNCTIONS. 

XXXV.        .     .     .    pp.  220,  229,  378,  397. 


m=oo 


w  =  l 

m  =  oo  (2m  +  l)2 


2  q  cos 

m=0 


m=l 


XXXVI p.  230. 

(1  -  2g2w-1cos27m  +  g4w~2), 

sin  TTW  JJ  (1  -  2  g2m  cos  2xu  +  g4m), 
2  Qo^  cos  TLU  JJ  (1  +  2  q2m  cos  2  TTW  +  g4m), 

m  =  l 
m  =  oo 


XXXVII p.  396. 

m  =  oo  m-oo 

O- n  u  -  92m>'       Qi=n(i  +  22'n>' 

m  =  l  wi"1 

TO  =  oo  m  =  oo 

Q2=  n  a  +  ?2m-i)>      ^=  n  (i  -  52m-1)- 

w  =  l  w»-l 

QiOaQs-1,  16gQi8=Q28-Q38.      •    .pp.  396,  409. 


TABLE   OF   FORMULAS  477 


XXXIX. 

If  v  =  — ,        z  =  e™         r  -- 


2      mti  (1  -  h2mz~2)2  (l  -h2^z2)2Y 

p.  336. 


p.  337, 

nVrS-  p-341' 

m  =  l  i 


P.342. 


-  yj  1  +  q2mz~2  "L-j.  1  +  ^2mg2  (  342, 

11      1  +  q2m     11    1  +52m    '       PP*  1 379. 


2m-l~-2 


_  - 

TT      __  -     _   TJ 

11        I+y—  1        11      l 


au  = 
3 


d+22" 

2yf  l-92m-lg-2»yf  j__ 
JLl         !_g2m-l       JL=1       1 


wz=»  ^          ^     2m-l  O 

e2*^  TT          ?  d"-?— " 


_  02m-l 
" 


478 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


TO=0 


#o(0)  =  1  +  2 


m=0 


+2 


=  0 


XL 

(2m  +  l)2 


pp.  397,  400. 


-  2q 


XLI  .........     See  p.  397. 

(l),  #i'd)  =  -  #i'(0). 


XLII  .....     See  pp.  397,  411. 


#O'(T)=  2^g-1#o(0),      #0'(w 
#i/W=-2-1#i'(0),       #i/(m 


#3'  (r)  =  -  2  wg  -  !  ^3  (0)  ,    #3'  (m  +  WT)  =  - 
#i'(0)  =  2  *Qo82*,  #o(0) 


~w2^3  (0)  , 

,  •     pp.  397,  399. 


TABLE   OF   FORMULAS. 


479 


OO)  = 


XLIII pp.  385,  410. 


-  e3 


e2—  63 


OO)'  = 


XLIV See  p.  410. 


-    Co 


w' 


QS       f 


6Z 


480 


THEORY    OF   ELLIPTIC    FUNCTIONS. 


XLV. 


p.  399. 


, 
h 


,QS  , 
•  pp-  398>  4i 


XLVI. 


p.  400. 


„ 
•     P- 244. 


=  4-00 


,  PJX400,  403. 


>2m2+w 


TABLE   OF    FORMULAS. 


481 


XL  VII. 


G  =  (61-  62)2fe-  e3)2(63- 


•i-        »t          f\     04.    O  >i  r\r\ 

=  Tc  ~T9~^o    5  •>    •     P-  409. 
16  to12 


i         { 408, 
'PP-    397. 


7T2 


12 


p.  408. 


2  to 


p.  409. 


03= 


4- 


—  63 


482 


THEORY   OF   ELLIPTIC   FUNCTIONS. 


CO 


XL  VIII  .........     p.  409. 

u  =  2cov. 


XLIX 
i'(0) 


2H  +l_^l 
C         w       2  co  &i( 


pp.  409,  304,  378. 


L 


p.  409. 


TABLE   OF   FOKMULAS. 


483 


LI. 


Vei—  63+  Vei  — e2 


=/i. 

^•^•MB 

•     •    '), 

p.  408. 

(l+2g4  +  231«+.  •  •), 
p.  409. 


(1  +g2w) 

=1 


2w2 


m  =  l 

77l  =  X 


2 pu 


p.  336. 
p.  379. 


^    1_38g1.2+58g2.3_ 


m  =  °°  r,2m  m  =  °°          «2m-l  w-00 


—  1 


V— f*" 

m  =  l 


p.  379. 


484 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


LIL 


[Formulas  (D),  p.  237]. 


LIIL 

022U  —   °32U  +  (02  —  63)  (72U  =  0, 

o32u  -  ai2u  +  (e3-  ei)o2u  =  0, 

oi2u  —  o22u  +(ei—  e2)o2u  =  0. 

(e2  —  e3)oi2u  +  (e3  —  ei)o22u  +  (e\  — 


p.  381. 


=  0. 


LIV.       .     ....     ."    .     pp.  305,  383,  387. 


(•  /  7N 

ou               1               (\/                       l,\ 

o\u  /            cn  (v  e  \  —  e3  •  u,  k) 

o3u     Vei-es 

ou                     sn(Vei—e3'U,k) 

G\W       rrt(\./f>         f>        11    If} 

&2U  \/  p         r    ct/i^V  t1]  —  ^3  •  'U,  k) 

"*u 

ou                     sn  (\^e  i  —  e3  •  u,  k) 

O2U  i      (\/f>     f>         11     If} 

&3W'  \/f         f> 

aau 

ou                     sn(\/ei—e3'U,k) 

o\u  •    CQom  (\/g   g    -  u  k] 

ou            1 

02U 

/  multiplied  by 
o\u     Ve\—e3 

o(u}             l 

tg  am  (v  e  i  —  e3  •  u,  k), 

02U                                       1 

,            multiplied  by 

o\u     sin  coam^ej  —  e3  •  u,  k) 

cos  coam  (\/e  i  —  e3  •  u,k), 

^3  \^/)       ^^^  ^  1  —  ^*^            1^.*     1'^ll-w 

03U                                    1 

—  •  multiplied  DV 

o\u     cos  am(v/ei  —  e3  •  u,  k) 

A  coair^Vei  —e3»u,k)} 

[Schwarz,  Zoc.  ci^.,  p.  30.] 


TABLE   OF   FORMULAS.  485 


LV. 
Homogeneity  ........     p.  343. 

Xco,  Xcof)  =  Xo(u,  CD,  cof), 


fa),  fa)')  =  i  £(u,  (o,  oj'), 


,  fa)')  =  —  @(u,  co,  a)'), 


^(u,  co,  co'), 


(*"' 


486  THEORY   OF  ELLIPTIC   FUNCTIONS. 

LVI  ..........     p.  252. 

sn"u  =  -  (1  +  k2)sn  u  +  2  k2snsu, 
cn"u  =  (2k2~  l)cnu  -  2k2cn3u, 
dri'u  =  (2  -  k2)dn  u-2  dn3u-, 

(1  +  14k2+  k4)snu  -  20k2(l  +  k2)sn*u  +  24k*sn5u, 
(1  -  16/b2+  16A;4)c/iw  +  20A;2(1  -  2k2)cn3u  +  24  A;4cnX 
(16  _  16  A;2  +  k*)dnu  +  20  (k2-  2)dn*u  +  24  dn5u. 
(See  also  Formulas  II.) 

LVII  ........     p.  252,  et  seq. 


, 

5! 

sn'(0)=  1,  -  sn'"(0)=  1  +  k2,  sn^(0}=  1  +  14A:2+  k4, 
sn<7>(0)=  1  +  135  k2  +  135  /b4+  A;6, 
sri(9)(0)=  i  +  1228  A;2  +  5478  k4  +  1228  kQ+  k8, 


cn"(0)=  -  l,cnW(0)=  1  +  4fc2,  -  cn^(0)-  1  +  44  fc2  +  16  fc* 
cnW(0)=  1  +  408  &2+  912  A;4  +  64  /b6, 
cn<10>(0)=  1  +  3688  k2  +  30768  /b4+  15808  kQ+  256  A;8, 


=  k2(k2  +  4),  -  dn<6)(0)=  k2(k*  +  44k2  +  16), 
dn<8HO)  =  k2(k&+  408  A;4  +  912  k2  +  64), 
dn<10>(0)=  A;2(A;8+  3688  kQ+  30768  &4  +  15808  A;2  +  256), 

[Gudermann,  CreUe,  Bd.  XIX,  p.  80.] 

kK      2  Ku      Vg  sinw    ,   Vo3  sin  3u    .    Vg5  sin  5w  , 

—  sn =  — +  — 5 —  +  — * = h  •  •  •  ,      T 

2;r          n  1—5  1  —  q3  1  —  q5 

kK      2  Ku  _  \/~q  cost£       \/q3  cos  3u       Vq5  cos  5u 

—  en  —     -   ~—        H        -  —    g       -T       :  ~    ^        i   *  *  *  > 

K   i    2  Kw  =  1    ,    q  cos  2^       g2  cos  4u    ,    q3  cos  6^ 

2^      ~^~  "4^     1  +g2  1  +g4  1  +  g6  '  ' 


TABLE   OF   FORMULAS.  487 

LVIII. 

Of  O2  .  Q  P\ f  O2      £\  O  -  P\ 

O I  ^        O          O I  ^        O  ^      O 

1       /oxx    v  e  ^2     3  5  ^3       o        I  9^2  i_        H^2^3 

9!^    W~^t~22^??    ~  22  •  7  ^  1        23.3-5^      24-3-5-7* 

[See  Art.  377.] 


LIX. 

+  •  •  •  4-  CnM2f|-2+  •  •      •  •        -  326-8. 


25  •  3  •  53  •  13      24  •  72  •  13 

•  -  p.  327. 


[n>3] 

LX. 

™  =  u-^u5-2dl^u7-'--  -  p-328- 

O  (J  -L  o  oi^o 

— • QtiUrO.      ...      p.  6\l6. 


1  -  ieAM2  -  -L  (6  e?-  g2)u*-  -  -  -  .     .    p.  394. 

-I  4o 

U  =  1,  2,  3.) 


488  THEORY   OF   ELLIPTIC    FUNCTIONS. 


LXI  ........    pp.  236,  246. 

u)&*a  +  i(v)-&a+i(u)#12(v), 

(a-  1,2,3;  #4=  00). 


p.  237. 


TABLE   OF   FOEMULA&  489 


LXII. 

o(u  +  u\)a(u  —  Ui)0(ii2  +  ^3)^(^2  —  ^3), 
+  0(u  +  112)  a  (u  —  U2)0(uz  +  u\)a(u^  —  HI), 
+  0(u  +  us)0(u  —  u3)  (7  (HI  +  u2)0(ui  —  w2)  =  0.  .  .  p.  390. 


a(u  +  v)0(u  -  v}  =  o2ua)2v  —  o?uo2v,    .....     p.  391 
(ev  —  etl)(j(u  +  v)a(u  —  v)=  o^ua.^v  —  a£ua,?v, 

o(u  +  v)0i(u  —  v)=  oruapv  -  (en  -  e^(ex-  ev)a2ua2v, 
(ev—  ej  01(11  -f  v)oi(u  -  v)  =  (ex—  e^ov2uov2v  —(e^-  e^ofuofv, 
V)GX(U  —  v)  =  o^ua^v  —  (ei  -  e^a2uov2Vj 


v)  a  (u  —  v)  =  o\u  ou  (7^  ovv  —  o^u 

0(U  +  V)  0i(u  —  v)  =  (JxU(7U  OpV  avV  +  0ftU  0VU  Otf)  0V, 

0ft(u  +  v)0i(u  —  v)  =  oiu  0ftu  0)V  0^  —  (eu  —  e\)  au  0vu  0v  0vv. 
.  p,  v  =  1,2,3.]  [Schwarz,  loc.  tit.,  p.  51.] 


490  THEOEY   OF   ELLIPTIC   FUNCTIONS. 


LXIIL       .     .     .      (See  pp.  273,  349,  364.) 

sn(u  ±  v)  =  (snucnv  dnv  ±  snvcnu  dn u)  -r-  D, 
where  D  =  1  —  k2sn2u  sn2v. 

cn(u  ±  v)  =  (en  ucnv  T  sn  u  sn  v  dn  u  dn  v)  -r-  D, 
dw(w  ±  v)  =  (dnudnv  T  k'2snu  snv  cnucnv)  +  D, 
sn(u  +  v)  +  S7i(i£  —  v)  =  2  sn  ucnv  dnv  -r-  D, 

+  v)  —  sn(u  —  v)  =  2  sn  v  en  u  dn  u  -f-  D, 

+  v)  sn(w  —  v)  =  (sn2u  —  sn2v)  -r-  D, 

+  v)  +  cn(w  —  v)=  2  cnucnv  -5-  D, 

—  v)  —  cri(w  +  v)  =  2  snudnusnv  dnv  -r-  D, 
+  v)  +  d?i(^  —  v)  =  2 dnudnv  -r-  D, 

—  v)  —  dw(w  +  v)=  2k2  snucnu  snv  cnv  -=-  D, 
k2sn(u  +  v)  sn(w  —  v)  =  (dn2v  +  k2sn2u  cn2v)  -f-  Z), 
sn(u  +  v)  sn(u  —  v)  =  (cn2v  +  s?i2it  dn2v)  -f-  D, 


+  dn(u  +  v)  dn(u  —  v)  =  (dn2u  +  dn2v)  -f-  Z), 

—  k2sn(u  +  v)  sn(u  —  v)  =  (dn2u  +  k2sn2v  cn2u)  -*-  Z>, 

—  sn(u  +  v)  sn(w.  —  v)  =  (cn2u  +  sn2v  dn2u)  -f-  D, 

—  cn(w  +  v)  cn(u  —  v)=  sn2u  dn2v  +  sn*v  dn2u  -r-  Z), 

—  dw(w  4-  v)  rf?z(w  —  v)  =  k2(sn2u  cn2v  +  sn2/y  cn2w)  -H  D. 


TABLE   OF   FORMULAS. 


491 


LXIV. 


02U02V 


r(u  +  v)+  r(u  -  v)-  2ru 

r(u  +  v)-£(u-v)-2  £v 


2    %>u  —  %>v 


=         , 


(6  y2u  -  ^ 


4  ?3?^  — 


p.  352. 
p.  352. 


p.352. 


).  353. 


pp.  366,  367. 


p.  355. 


492  THEORY   OF   ELLIPTIC    FUNCTIONS. 


LXIII  (Continued). 

{  1  ±     sn(u  +  v)  }  {l  ±     sn(u  —  v)  }  =  (cnv  ±     snu  dnv)2  +  D, 
{l±     sn(u  +  v)  }   JIT     sn(u  —  v)  }  =  (cnu  ±     snv  dnu)2  ^-D, 
{  1  ±  k  sn(u  +  v)  }   {  1  ±  k  sn(u  —  v)  }  =  (dn  v  ±  ksnucn  v)2-+-  D, 
{  1  ±  k  sn(u  +  v)  }   {  1  T  A;  sri(w  —  v)  }  =  (dnu  ±  k  snv  cnu)2-^  D, 
\  I  ±     cn(u  +  v)  }   {  1  ±     cn(u  —  v)\  =  (cnu  ±  en  v)2  +  D, 
\  1  ±     cn(u  +  v)  }  {  1  -F     cn(u  —  v)  }  =  (snu  dnv  =F  STIV  dnu)2+-  D, 
jl  ±     rfn(^  +  v)  }  {l  ±    dn(u  -  v)  }  =(dnu  ±  dnv)2+  D, 
1  ±     dn(u  +  v)       1  ~F     dn(w  —  v)    =  k2(snu  cnv  T  swv  cnu)2-r-  D. 


sn(u  +  v)  cn(^  —  f  )  =  (snu  cnu  dnv  +  snv  cnv  dnu}  ~  D, 
sn(u  —  v)  cn(u  +  v)  =  (snu  cnu  dnv  —  snv  cnv  dnu)  +  D, 
sn(u  +  v)dn(u  —  v)  =  (snudnu  cnv  +  snv  dnv  cnu)^-  D, 
sn(u  —  v)dn(u  +  v)  =  (snudnu  cnv  —  snv  dnv  cnu)-r-  D, 
cn(u  +  v)dn(u  —  v)  =  (cnu  dnu  cnv  dnv  —  k'2  snu  snv)  +  D, 
cn(u  —  v)dn(u  +  v)  =  (cnu  dnu  cnv  dnv  +  k'2  snu  snv)  -r-  D. 

sin  {  am(w  +  v)  +  am(w  —  v)  J  =  2  snu  cnu  dnv  -5-  D, 
sin  |am(&  +  v)—  am(w  —  v)  }  =  2  snv  cnv  dnu--  D, 
cos  {  am(w  +  v)  +  am  (it  —  v)  }  =  (cn2u  —  sn2u  dn2v)  -=-  D, 


cos  {  am(w  4-  v)  -am  (w  —  v)  }  =  (cn2v  -  sn2v  dn2u)  -=-  D. 

(Jacobi,  Werke,  I,  pp.  83-85.) 


TABLE   OF   FORMULAS. 


LXV. 


.     -     p.  353. 


d2 
2%>u  -  —  log 

d2 


-  gw), 


-f  f)  -  <?(tt  -  t;)  =  - 


log  (gni  -  gw), 


.  354< 


0 p.  354. 


t        ^      .     . 
4 


=  2  ^((7^3-  3 


w,  p.  356. 


(7(2  U)  =  2 


.....      .....       p.  380. 

(Schwarz,  /oc.  c^.,  p.  14.) 


494  THEOKY   OF   ELLIPTIC   FUNCTIONS. 

LXVI. 

rr$  / : — 
dn2udu  =  I    V  1  —  k2  sin2  (f>  d<j)  =  E((f>,k),    ....     p.  285. 
«/o 


=    fVl  -k2  sin2  0d0,    £7'=  fVl  -  fc/2  sin2  0  d0. 
Jo  Jo 

KE'  +  K'E-KK'=  -,        .....     p.  291. 
2 

J  =  K-E,     J'=E';    J'K-K'J  =  f- 

2i 

®(u)=®(0)eS"Z(u}du  .......    p.  292. 

j-}du  =  E(u)-u 
K/  K 


-  |V-  r 
.     K/         Jo 


Z'(0)  =  1  -  f 
/v 


=  Z'(0)  -  Z'(tt),  /b2cn2^  =  k2-  Z'(0)  +  Z'(w), 


Z(0)=0, 
-  f ,  .    .    p.  294. 


,  .     .     p.  292. 


tan  am(u,  k')dn(u,  k')  +     ^     +  Z(w,  A;').      p.  293. 
2  KK 


TABLE   OF   FORMULAS.  495 


LXVII  .....     ,    .    .  pp.  302-303. 

E  --  ^—  K  I  ,     T/  =  -  iV^^73  \  Ef  +  —^—Kf  I  > 
ei  -  «3      J  (          el  -  e3       ) 


Vei-  e*  Vei  -  e3 

Formulas  for  £u  are  found  under  Nos.  XXII  and  XLIV. 


u  =  ^ 

e\-  e3-u)  =  ^  /    1         ^^+eiM],    .     B     B          p.  307. 

'!•     p.  308. 

p.  295. 


v)  =  Z(M)  +  Z(v) -  fc2sn z^ snvsn(u  +  v),     ...     p.  350. 


496 


THEORY   OF   ELLIPTIC    FUNCTIONS. 


LXVIII. 


n[z, 

i  p 

2J 


f  \Z(ai);  «2,  v^Z(a2)=  H(z;  au 


z  — 


\/Z(z) 


414 


Z(Z)=    (1   - 


2  ,vA, 

(1  +  nsm2</>)A<£ 


n(     fl)  = 


+  a)          0  (a) 


p.  420. 
p.  420. 


II(M,  a)-  II(a,  u)=  uE(a)-  aE(u),     .....     p.  421 

n(w,o)=-H(-  w,o),     H(0,a)=  0=n(w,K), 
w,  iK')  =  oo,     H(K7  a)  -  KE(a)  -  aE  =  KZ(a), 


U(u  +  2K,a)=  H(u,a)+  2K 


K 


U(iu,ia  +  K)=  U(u,a  +  K',k'), 


Q/  /2  K 
2jCi/       y    TT 


e/2Ka\ 
Addition-theorems  are  found  on  p.  426. 


p.  421. 
p.  422. 

p.  423. 


TABLE   OF   FORMULAS.  497 

LXIX. 

,  \AS~CO;  a,  VS(a);   oc)  =  U(t;  a;  oo) 


dt       .„,,       3 

'     S®=**-92t-to,    .     p.  419. 


t  =  &u,         VS(t)  =  -ff'u,         a  = 

I'u  +  o'\ 


;  a;  oc)  -  II(a;  «;  oc)  =  u  -  UQ         +  (2n  +  l)m.  .     p.  420. 

CT^o  (7W 

Addition-theorem  on  p.  429. 

LXX. 
J   E(u}du  =  logfl(w),  ........     p.  423. 

_M* 

Q(iu)=e    2cn(u,k')n(u.  k'),  .     .     .     .     p.  424. 
(Vei-  e3-u)=  e^e^o3u,  ........     p.  425. 

n(M,a)=Mjg(a)+llogQ(M~g),       .     .     p.  424. 
2        O(i^  +  a) 


^73  .  u,  V^T^  .  a)  =  1  ^  gs(»  -  a)  +  u        , 

2        (73  (u  4-  a)          <73a 


498  THEORY   OF   ELLIPTIC   FUNCTIONS. 

LXXI. 

If  f(u)  is  a  rational  function  of  u,  we  may  write 

(l)  /(«)-  A  .f-V+V'  ----  ± 


where  vt-  =  -  ........     p.  9. 


u  — 


If  <£(w)  is  a  rational  function  of  sin  u  and  cos  w,  we  may  write 

(1)  <f>(u)=  P(eiu)  +  3>(u),      .    "...     .     .     p.  22. 

where 

*./  \       75  o         ,u  —  di      B2{  d         u  —  di 

<S>(u)=B+BliCOi 


vf 
%\ 

i        u-^1 
f0         2     J 


B3i  d*        u-_a<  Bnti 

"C<     ~~  : 


(2)     ^(u)=  C^  "  "      .  '  •  •  "  .".     -     -     P-  25. 

sm  (it  —  61)  sin  (u  —  62)   •  •  •  sin  (u  —  bn) 


If  F(w)  is  a  doubly  periodic  function,  we  may  write 
(1)  F(u)=  D 

±  Zo^-^Cw  -  ^),     .     .     .    pp.  120  and  433. 

where  the  transcendental  function  Z0(w)  becomes  infinite  of  the  first  order 
for  u  =  0,  the  residue  being  unity. 


(2)      FfrHC-  ••••  "/"  ~Mf),  .     p.  439. 

<T(M  —  UI)<T(U  —  u2)   .   .   .    <T(M  —  Ut>) 

where  WIQ+  w2°  +  •  •  •  +  ur°=  ui+  u2+  -  •  -  -\-  ur. 


7  7n  o 


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